On conciseness and profinite RAAGs

PhD Thesis On Conciseness and Profinite RAAGs Matteo Pintonello Supervisors: Montserrat Casals-Ruiz Gustavo A. Fernández-Alcober December 2023 (cc) 2023 Matteo Pintonello (cc by-nc 4.0) This thesis has been carried out at the University of the Basque Country (UPV/EHU) under the financial support of the grant FPI-2018 of the Spanish Government. In addition, the author was supported by the Basque Government, projects IT974-16 and IT483-22, and the Spanish Government, projects MTM2017-86802-P and PID2020-117281GB-I00, partly with ERDF funds. ii Contents Acknowledgments vii Introduction 1 Resumen de la tesis en castellano 7 1 Problems on group words 13 1.1 Words and verbal subgroups . . . . . . . . . . . . . . . . . . . . . 14 1.2 Varieties and relatively free groups . . . . . . . . . . . . . . . . . 15 1.3 On three questions of P.Hall . . . . . . . . . . . . . . . . . . . . . 17 1.4 Conciseness .............................. 18 1.5 Conciseness in other classes of groups . . . . . . . . . . . . . . . . 20 1.6 A comprehensive list of known concise words . . . . . . . . . . . . 22 1.7 Tables of concise words . . . . . . . . . . . . . . . . . . . . . . . . 26 2 Conciseness on normal subgroups 29 2.1 Preliminaries ............................. 30 2.2 Lower central words . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 An example: the word δ2....................... 37 2.4 Outer commutator words . . . . . . . . . . . . . . . . . . . . . . . 41 3 Counterexamples to Hall’s conjecture 51 3.1 Elements of small cancellation theory . . . . . . . . . . . . . . . . 52 3.2 Ivanov’s counterexample . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 Olshanskii’s counterexample . . . . . . . . . . . . . . . . . . . . . 62 3.4 Olshanskii’s word in profinite groups . . . . . . . . . . . . . . . . 65 3.5 On generation of verbal subgroups . . . . . . . . . . . . . . . . . 68 4 Coprime commutators 71 iii 4.1 History of coprime commutators . . . . . . . . . . . . . . . . . . . 72 4.2 Preliminaries ............................. 74 4.3 Introduction to the proofs . . . . . . . . . . . . . . . . . . . . . . 77 4.4 The meta-pronilpotent case for γ∗ k.................. 81 4.5 The poly-pronilpotent case for δ∗ k.................. 83 4.6 Strong conciseness of coprime commutators . . . . . . . . . . . . 94 5 Profinite right-angled Artin groups 97 5.1 Profinite groups acting on profinite trees . . . . . . . . . . . . . . 98 5.2 Basics on pro-CRAAGs ....................... 103 5.3 Direct product decomposition of pro-CRAAGs . . . . . . . . . . 106 5.4 Centralisers and normalisers of elements . . . . . . . . . . . . . . 109 5.5 Subgroups of pro-Cand pro-pRAAGs ............... 111 6 Abelian splittings of RAAGs 117 6.1 Abelian splittings of profinite RAAGs . . . . . . . . . . . . . . . . 118 6.2 JSJ decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.3 (A,H)-JSJ decomposition of pro-CRAAGs . . . . . . . . . . . . 124 6.4 A-JSJ decomposition of pro-CRAAGs ............... 127 Bibliography 139 iv To my parents. v vi Acknowledgments A PhD is not only an academic achievement, but a stage of life in which we have both great and hard times. Looking back at my time in Bilbao, I am overwhelmed by the amount of people that have helped me during these years. First of all, the two people without whom I would not be here, my advisors. Gustavo, thank you for wanting me here, for helping me and pushing me, and for always being positive. I cannot say I’m a full believer yet, but you taught, and showed me with facts, that it’s better not to be a non-believer. You were right, and I owe you a lot! To Montse, I cannot put into words how thankful I am for all the help you gave me! You stepped up in the moments I was struggling, you supported me a lot, and you have always been patient. Among all the great and amazing people I met here in Bilbao, you are the kindest one! Apart from my advisors, I’ll forever be grateful to Ilya for supporting me, not only economically. Half of the mathematicians that I met in Bilbao were here thanks to you, you must be proud of the network you are creating! I thank every member of the committee that gracefully accepted to read and correct this thesis: Marta, Enric, and Leire. Thanks also to all the substitute members, in particular to Eloisa for writing the first report. It is undeniable the importance that my coauthors have had in this work, each one of them was essential in my PhD. First of all to Pavel Zalesskii, thanks for always answering all my silly questions with your enormous knowledge and experience, and for all your patience. Then, a gigantic thank you to Pavel Shumyatsky. Your positive attitude really inspired me, and I’m grateful for collaborating with a mathematician I deeply admire. Iker, we went through a lot of adventures (and disadventures) together. I really owe you a lot, thanks for the motivation that you gave me, and for always believing! I also deeply thank Cristina for the opportunity of visiting her in Modena. In these years in Bilbao I met many important people in my life, and I can’t describe how happy I am for having had them in my life. I’ll try to thank them all, roughly in chronological order. First of all Marialaura: I will be in debt vii that this same word is strongly concise in profinite groups, settling that these problems differ substantially from the classical questions in abstract groups. Then, the thesis pursues the study of problems in profinite groups, beginning from some results related to strong conciseness. As we remarked, this problem could be split into two different sub-problems: proving that if |w{G}| <2ℵ0for a word win a group G, then w{G}is finite, and then proving conciseness in residually finite groups for w. For this reason, several results on strongly concise words relied on the additional hypothesis that, if a verbal subgroup of a profinite group is topologically finitely generated, then it can be generated by finitely many word values. We provide an example, with lower central words, that shows that this additional condition is not always satisfied. We then study strong conciseness for higher order coprime commutators, that are maps strongly resembling group words. They are a useful tool to generate some important characteristic subgroups of profinite groups, like pronilpotent residuals, with an accurately chosen generating set. Similarly to usual words, we can ask whether they are (strongly) concise, in the sense that in any group with finitely many (or less than 2ℵ0) coprime commutators, these elements generate a finite subgroup. It was shown by Acciarri, Shumyatsky and Thillaisundaram that higher order coprime commutators are concise in residually finite groups, while Detomi, Morigi and Shumyatsky proved that the basic coprime commutator map γ∗ 2is strongly concise. In a joint work with de las Heras and Shumyatsky, the author proved in [39] that higher order coprime commutators γ∗ kand δ∗ kare strongly concise in profinite groups, and we provide a full detailed proof of these results. In the second part of the thesis, we initiate the study of profinite right angled Artin groups. Abstract right angled Artin groups (RAAGs) are finitely generated groups whose only relations are commutators in the generators. These groups have a finite graph associated to their presentation, and they include, among others, free groups, free abelian groups and free or direct products of them. The central idea in geometric group theory is to study groups via actions on spaces. For example, free action of groups on a space should provide a connection between the geometry of the space and the algebra of the group. This is the case with actions on trees: a group acts freely if and only if the group is free. If we do not require the action to be free, Bass-Serre theory gives a description of the structure of groups acting on trees through HNN extensions and amalgamated products. If, rather than on a single tree, we require our group Gto act on a direct product of two trees, then the situation is different. Indeed Burger and Mozes constructed infinite simple groups acting freely and cocompactly on them. However, Bridson, 4 Howie, Miller and Short proved that if we require some additional residual properties, then such a group Gis virtually a direct product of free groups. These results were generalised by Haglund and Wise who proved that groups acting freely, and with some additional conditions, on CAT(0) cube complexes are subgroups of RAAGs. As profinite groups satisfy good residual properties, one can asked if no further conditions are required in this setting, namely a profinite group acts on a direct product of two profinite trees (or, even more ambitiously, on a profinite cubing) if and only if it is virtually a subgroup of a profinite RAAG. In order to approach this line of research, we must first study systematically profinite RAAGs. For a generic pseudovariety Cof finite groups, pro-CRAAGs are the pro-Ccompletion of abstract RAAGs and have been studied by Wilkes, Kropholler, Snopce and Zalesskii. In accordance to the contents of the article [16], joint with Casals-Ruiz and Zalesskii and currently in preparation, we study pro-CRAAGs using profinite Bass-Serre theory as the main tool. This theory is an analogue of the abstract one developed mainly by Mel’nikov, Ribes and Zalesskii. We use these methods to obtain standard properties of pro-CRAAGs, like the structure of their centralizers, studying a Tits alternative for their subgroups, and characterizing 2-generated subgroups of pro-pRAAGs. We then describe some properties of a pro-CRAAG that are immediately detectable by studying their underlying graph. For example, Krophopller and Wilkes already observed that a profinite RAAG splits as a free product if and only if the underlying graph is disconnected. We prove that pro-CRAAGs are directly decomposable if and only if their underlying graph is a join, and we then obtain a characterization of their splittings, as pro-Camalgams or HNN extensions, over abelian subgroups. We then continue the investigation of their abelian splittings by defining JSJ decompositions. These constructions are a description of all the ways a group G can split over a certain class Aof subgroups, and they can be either general (so A-JSJ decompositions) or relative to another class Hof subgroups (the so-called (A,H)-JSJ decompositions), in the sense that we require all the subgroups of G in the class Hto be elliptic. We give a constructive proof of the existence of the (A,H)-JSJ decomposition of a pro-CRAAG Gchoosing Ato be the class of abelian subgroups, and with the assumption that canonical generators of Gact elliptically. We then conclude by obtaining the general A-JSJ decomposition of the pro-CRAAG G. 5 Structure of the Thesis In Chapter 1 we give an overview of the known theory of conciseness, giving a considerable importance to the historical development of the theory. In Chapter 2 we prove that outer commutator words are concise on normal subgroups. This will be obtained first in the case of lower central words, and we will then approach the general proof by giving an explicit description for w=δ2, and then concluding with the proof of the general case. Chapter 3 will be devoted to the description of the counterexamples on conciseness, and then to the proof that Olshanskii’s word is boundedly concise in residually finite groups and strongly concise in profinite groups. We conclude the chapter giving an example of a profinite group with procyclic derived subgroup, but whose subgroup cannot be generated by finitely many commutators. In Chapter 4 we prove that higher order coprime commutators γ∗ kand δ∗ kare strongly concise in profinite groups. In Chapter 5, after an overview of profinite Bass-Serre theory, we focus on proving basic properties of profinite RAAGs, like the structure of their centralizers, and on characterizing their abelian splittings. We conclude the investigation of their abelian splittings in Chapter 6, where we explicitly construct their general and relative abelian JSJ decompositions. 6 Resumen de la tesis en castellano Aunque el origen de la teoría de grupos suele atribuirse a los trabajos de Galois, Jordan y Klein, todos estos trabajos estuvieron motivados por la conexión que esta disciplina tiene con la teoría de números o con la geometría. La teoría de grupos abstractos discretos obtuvo un interés independiente, sin inspiración geométrica, principalmente a principios del siglo XX, y un hito para ello se debe a los trabajos de William Burnside. En 1902 él preguntó si un grupo de torsión finitamente generado es necesariamente finito [15], actualmente nos referimos a esta cuestión como el “Problema de Burnside”. Este trabajo despertó el interés por problemas aún más profundos, como el estudio de la finitud de los grupos finitamente generados de exponente finito, también llamado “Problema de Burnside acotado”. Explícitamente, Grün [36] se preguntó si un grupo Gfinitamente generado que satisface gn= 1 para todo g∈Ges necesariamente finito. Podríamos observar que este problema se puede encuadrar en el contexto más amplio de una de las preguntas más naturales que se pueden hacer sobre una estructura algebraica, que es “¿Qué podemos decir sobre un grupo si este grupo sigue una regla fija?” Por supuesto, la pregunta es extremadamente heurística, pero podemos ver muchos de los primeros resultados en teoría de grupos a través de este enfoque, que puede ser interpretado como un ejemplo de un problema de palabras en grupos. Una palabra de grupo wes una concatenación finita de variables y de sus inversas, que puede verse como un elemento del grupo libre generado por nvariables 7 x1, . . . , xn. Para cualquier grupo G, la palabra wdefine naturalmente una aplicación de GnaG, simplemente sustituyendo los elementos del grupo en las variables de todas las formas posibles. La imagen de esta aplicación es el conjunto de valores de la palabra en G, normalmente denotado por w{G}, y el subgrupo w(G)que generan se llama subgrupo verbal. De especial interés es el estudio de las variedades de grupos, que son las clases de grupos en las que una determinada palabra wes una ley, en el sentido de que toma sólo el valor trivial. Las “reglas”mencionadas en la pregunta de Grün son simplemente leyes en el grupo, así que en términos modernos la cuestión es cómo estudiar la variedad de grupos generada por la ley xn. Otros problemas que pueden verse desde esta óptica son el estudio de los grupos abelianos, nilpotentes y resolubles de clase acotada, que son las variedades generadas por la palabra conmutador [x1, x2], por una palabra central inferior o por una palabra derivada respectivamente. En lugar de estudiar sólo los grupos en los que una palabra wes una ley, también podríamos preguntarnos si el hecho de que wtome un número finito de valores en un grupo Gtiene alguna implicación en la estructura de G. Es fácil darse cuenta de que cualquier grupo con un número finito de conmutadores es finito-por-abeliano, o, en otras palabras, si el conjunto de valores de γ2es finito en un grupo G, entonces el subgrupo verbal correspondiente es finito. Philip Hall se dio cuenta de que lo mismo es cierto para todas las palabras potencia xn, y las centrales inferiores γk, no sólo para γ2. Como consecuencia, Hall conjeturó que para cualquier palabra de grupo, si el conjunto w{G}de valores en un cierto grupo Ges finito, entonces el subgrupo verbal w(G)también es finito. Si una palabra satisface esta propiedad para cualquier grupo G, se llama concisa y, si lo hace para todos los grupos de una clase dada C, se dice que es concisa en C. Se ha demostrado que muchas palabras son concisas, además Merzljakov demostró que todas las palabras son concisas en grupos lineales, pero Ivanov construyó un contraejemplo para el caso general utilizando la Teoría de la cancelación pequeña. Más tarde, Olshanskii y Storozhev obtuvieron otros contraejemplos con métodos similares. El estudio de las palabras concisas progresó de todos modos, tanto buscando nuevas palabras que fueran concisas en todos los grupos, como estudiando el mismo problema en otras clases de grupos. Como los grupos lineales finitamente generados son residualmente finitos, el candidato natural para la mayor clase de grupos en los que todas las palabras son concisas es la clase de los grupos residualmente finitos. Es interesante observar que una palabra es concisa en grupos residualmente finitos si y sólo si es concisa en grupos profinitos, por lo que recientemente se ha propuesto otro avance importante. Cada grupo profinito de cardinalidad menor que 2ℵ0es finito, y se sugirió que 8 un fenómeno similar ocurre también para los valores de las palabras, llevando a la conjetura de que cada conjunto de valores de palabras con menos de 2ℵ0valores es finito. Uniendo este problema abierto con la conjetura de que todas las palabras son concisas en grupos residualmente finitos, tiene sentido definir que una palabra es fuertemente concisa en grupos profinitos si, siempre que tome menos de 2ℵ0 valores, su subgrupo (cerrado) verbal es finito. En la primera parte de esta tesis se discuten varias contribuciones que el autor ha aportado a la teoría de los problemas de concisión. La primera contribución se refiere a la versión más general del problema, que consiste en buscar nuevas palabras concisas en todos los grupos. Una de las primeras clases de palabras que Philip Hall demostró que son concisas son las palabras no conmutadoras, es decir, las palabras que no se encuentran en el subgrupo derivado del grupo libre generado por las variables. Más recientemente, Delizia, Shumyatsky, Tortora y Tota demostraron que lo mismo es cierto para la palabra γ2(u1, u2), donde u1, u2son palabras no conmutadoras disjuntas (es decir, en conjuntos disjuntos de variables). Este resultado fue generalizado en 2022 por Azevedo y Shumyatsky, quienes demostraron que la palabra γ3(u1, u2, u3), para uipalabras no conmutadoras disjuntas, es concisa. En [34], Fernández-Alcober y el autor demostraron que w(u1, . . . , uk), con ui palabras no conmutadoras disjuntas, es concisa en el caso de que wsea una palabra central inferior (demostrando una conjetura de Azevedo y Shumyatsky), y en el caso de que wsea una palabra derivada. Los argumentos del artículo mencionado también funcionan, con algunas pequeñas modificaciones, cuando wes un conmutador externo, por lo que probamos completamente este caso, que incluye y generaliza el caso de las palabras centrales inferiores y derivadas. En realidad obtenemos una propiedad más fuerte y demostramos que todos los conmutadores externos son concisos en subgrupos normales, en el sentido de que siempre que el conjunto de valores que toma la palabra en una tupla Nde subgrupos normales sea finito, entonces el subgrupo que generan también lo es. Estas nuevas palabras concisas intentan acercarse al límite entre las palabras concisas y las que no lo son. De hecho, actualmente se desconocen condiciones generales para que una palabra no sea concisa. Las técnicas utilizadas para construir los tres contraejemplos conocidos, el de Ivanov, de Olshanskii y de Storozhev respectivamente, han sido desarrolladas dentro de la Teoría de la cancelación pequeña. Esta área de la teoría geométrica de grupos se basa en la idea de que, si las relaciones de una presentación fija G=hS|Ride un grupo satisfacen algunas condiciones adicionales, es posible deducir algunas propiedades geométricas y algebraicas de los grupos. Esto se consigue observando diagramas, construidos 9 utilizando las relaciones de G, que representan elementos triviales en el grupo. La construcción completa de las tres palabras no concisas, y de los grupos en los que estas palabras no son concisas, es bastante técnica. Por ello, sólo trataremos de dar una idea general del resultado de Ivanov y, a continuación, destacaremos algunas diferencias entre las tres palabras no concisas. A continuación, nos centramos en el contraejemplo de Olshanskii. Como demostraron Shumyatsky y el autor en [68], la palabra de Olshanskii, que no es concisa en general, es en realidad concisa en grupos residualmente finitos. Este es el primer ejemplo de una palabra que no es concisa en todos los grupos, pero es concisa en grupos residualmente finitos. Luego mostramos también que esta misma palabra es fuertemente concisa en grupos profinitos, estableciendo que estos problemas difieren sustancialmente de las cuestiones clásicas en grupos abstractos. La tesis prosigue con el estudio de problemas en grupos profinitos, partiendo de algunos resultados relacionados con la concisión fuerte. Como comentamos, este problema podría dividirse en dos subproblemas diferentes: probar que si |w{G}| <2ℵ0para una palabra wen un grupo G, entonces w{G}es finito, y luego probar la concisión en grupos residualmente finitos para w. Por esta razón, varios resultados sobre palabras fuertemente concisas se basaban en la hipótesis adicional de que, si un subgrupo verbal de un grupo profinito es topológicamente finitamente generado, entonces puede ser generado por un número finito de valores de la palabra. Aportamos un ejemplo, con palabras centrales inferiores, que muestra que esta condición adicional no siempre se cumple. A continuación, estudiamos la concisión fuerte para conmutadores coprimos de orden superior, que son aplicaciones muy similares a las palabras de grupo. Son una herramienta útil para generar algunos subgrupos característicos importantes de los grupos profinitos, como los residuales pronilpotentes, con un conjunto generador elegido con cuidado. De forma similar a las palabras usuales, podemos preguntarnos si son (fuertemente) concisas, en el sentido de que en cualquier grupo con un número finito (o menor que 2ℵ0) de conmutadores coprimos, estos elementos generan un subgrupo finito. Acciarri, Shumyatsky y Thillaisundaram demostraron que los conmutadores coprimos de orden superior son concisos en grupos residualmente finitos, mientras que Detomi, Morigi y Shumyatsky demostraron que el conmutador coprimo básico γ∗ 2es fuertemente conciso. En un trabajo conjunto con de las Heras y Shumyatsky, el autor demostró en [39] que los conmutadores coprimos de orden superior γ∗ kyδ∗ kson fuertemente concisos en grupos profinitos, y nosotros proporcionamos una demostración detallada completa de estos resultados. En la segunda parte de la tesis, iniciamos el estudio de los grupos de Artin de án10 gulos rectos profinitos. Los grupos abstractos de Artin de ángulos rectos (RAAGs) son grupos finitamente generados cuyas únicas relaciones son conmutadores en los generadores. Estos grupos tienen un grafo finito asociado a su presentación, e incluyen, entre otros, los grupos libres, los grupos abelianos libres y los productos libres o directos de ellos. La idea central de la teoría geométrica de grupos es estudiar los grupos mediante acciones en espacios. Por ejemplo, la acción libre de grupos en un espacio debería proporcionar una conexión entre la geometría del espacio y el álgebra del grupo. Éste es el caso de las acciones en árboles: un grupo actúa libremente en un árbol si y sólo si el grupo es libre. Si no exigimos que la acción sea libre, la teoría de Bass-Serre proporciona una descripción de la estructura de los grupos que actúan en árboles en términos de extensiones HNN y productos amalgamados. En lugar de en un único árbol, si requerimos que nuestro grupo Gactúe en un producto directo de dos árboles, entonces la situación es diferente. En efecto, Burger y Mozes construyeron grupos simples infinitos que actúan libre y cocompactamente en ellos. Sin embargo, Bridson, Howie, Miller y Short demostraron que si exigimos algunas propiedades residuales adicionales, entonces tal grupo G es virtualmente un producto directo de grupos libres. Estos resultados fueron generalizados por Haglund y Wise, quienes demostraron que los grupos que actúan libremente, y con algunas condiciones adicionales, en complejos cúbicos CAT(0) son subgrupos de los RAAG. Como los grupos profinitos satisfacen buenas propiedades residuales, cabe preguntarse si no se requieren más condiciones en este contexto, a saber, que un grupo profinito actúa en un producto directo de dos árboles profinitos (o, aún más ambicioso, en una cubicación profinita) si y sólo si es virtualmente un subgrupo de un RAAG profinito. Para abordar esta línea de investigación, primero debemos estudiar sistemáticamente los RAAG profinitos. Para una pseudovariedad genérica Cde grupos finitos, los RAAG pro-Cson la compleción pro-Cde los RAAG abstractos y han sido estudiados por Wilkes, Kropholler, Snopce y Zalesskii. De acuerdo con el contenido del artículo [16], conjunto con Casals-Ruiz y Zalesskii y actualmente en preparación, estudiamos RAAGs pro-Cutilizando la teoría profinita de Bass-Serre como herramienta principal. Esta teoría es un análogo de la abstracta desarrollada principalmente por Mel’nikov, Ribes y Zalesskii. Utilizaremos estos métodos para obtener propiedades estándar de los RAAGs pro-C, como la estructura de sus centralizadores, estudiando una alternativa de Tits para sus subgrupos, y caracterizando subgrupos 2-generados de RAAGs pro-p. A continuación, describiremos algunas propiedades de un RAAG pro-Cque se pueden detectar inmediatamente a partir de su grafo subyacente. Por ejemplo, 11 Krophopller y Wilkes ya observaron que un RAAG profinito se descompone como producto libre si y sólo si el grafo subyacente es disconexo. De manera dual, demostraremos que un RAAG pro-Cse descompone como producto directo si y sólo si su grafo subyacente es una suma de grafos, y a continuación obtendremos una caracterización de sus decomposiciones, como amalgamas pro-Co extensiones HNN, sobre subgrupos abelianos. Posteriormente, continuamos con la investigación de las decomposiciones abelianas de un RAAG pro-C, esta vez en el contexto de las decomposiciones JSJ. Estas construcciones son una descripción de todas las formas en que un grupo G puede decomponerse sobre una cierta clase Ade subgrupos, y pueden ser generales (por tanto descomposiciones A-JSJ) o relativas a otra clase Hde subgrupos (las llamadas descomposiciones (A,H)-JSJ), en el sentido de que requerimos que todos los subgrupos de Gen la clase Hsean elípticos. Daremos una prueba constructiva de la existencia de la decomposición (A,H)- JSJ de un RAAG Gpro-Celigiendo Acomo la clase de subgrupos abelianos, y con el supuesto de que los generadores canónicos de Gactúen elípticamente. Concluiremos obteniendo la descomposición general A-JSJ del pro-CRAAG G. 12 1 Problems on group words In this chapter we set the foundations of the theory of concise words. Initially we give the basic definitions of word maps and verbal subgroups. We then describe varieties of groups, that are one of the main motivations driving the development of word problems in groups. In Section 3, we give the formulation of three conjectures of Philip Hall. We briefly analyse the partial answer to the first two of them, and then we discuss the follow-up of the third problem in the fourth section. Indeed, the last question of Hall consisted in proving that, if a word wtakes finitely many values in a group G, the associated verbal subgroup is finite. A word satisfying this is said to be concise. We describe the partial positive answers and then mention the counterexamples to Hall conjecture. In Section 5, we describe the more recent driving areas in conciseness, namely the study of words that are concise in residually finite groups. A further investigation is due to the conjecture that every word wis strongly concise in profinite groups, meaning that whenever the set of w-values has less than 2ℵ0elements, then the verbal subgroup is finite. In Section 6 we glide over all the results of conciseness, addressing in which threads of investigation the mathematical community was able to make improvements, and then conclude with a summary of the best results obtained so far in each direction. 13 We can therefore restrict our search to commutator words. In [81], Turner-Smith proved that lower central words γkare concise (this was already known to P. Hall), moreover he extended the result to derived words δk, but the arguments in this case are already more advanced. For several years the problem was untouched, until Wilson proved in [82] that all outer commutator words are concise. The dreams of obtaining an affirmative answer to conciseness problems were shattered by a counterexample, obtained by Ivanov in 1989 [43]. We will give some ideas of the construction of this counterexample in Section 3.2. Still, many more words have been proved to be concise. Even further, many words have been proved to be boundedly concise. Definition 1.14 (Boundedly concise words).A word wis boundedly concise in a class Cof groups if there exists a function f:N→Nsuch that, if there is a group G∈ C with |w{G}| ≤ m, then |w(G)| ≤ f(m). In 2009, Fernández-Alcober and Morigi obtained a different proof of conciseness of outer commutator words in [31]. In the same article, there are two proofs of the following result, one by the authors and one that was communicated to them by Mann. Theorem 1.15. Any word wthat is concise is boundedly concise. 1.5 Conciseness in other classes of groups The counterexample of Ivanov did not impede pursuing better and further results on conciseness. In particular, a huge development of the theory shifted toward proving in which classes of groups every word is concise. Definition 1.16 (Verbal conciseness).We will say that a class of groups Cis verbally concise if, for every group G∈ C and any group word wwe have that, if w{G}is finite, then w(G)is finite too. Some classes of groups that are obviously verbally concise are abelian groups (because, if Gis abelian, w(G) = w{G}) or finite groups. Turner-Smith proved that each word is concise in residually finite groups such that all of their quotients are residually finite [81]. The most important open conjecture regarding conciseness is the following. This conjecture was discussed by several authors, but it is usually attributed to either Jaikin-Zapirain or Segal. Conjecture 1.17. The class of residually finite groups is verbally concise. 20 Studying conciseness in residually finite groups involves a different machinery compared to the analogous problem in general abstract groups. These additional tools made it possible to prove that some words, which are unknown to be concise or not in general, are actually concise in residually finite groups. Consider, as an example, Engel words, that are defined iteratively as e1(x, y)=[x, y]and en= [en−1, y] = [x, y, n . . ., y]. It is known that these words are concise only in the cases of n≤4(see [1] [32]), but it is unknown whether they are in general. However, all these words are concise in residually finite groups ([26]). Any residually finite group embeds in its profinite completion, so it is a natural question whether the study of conciseness in profinite groups can yield an affirmative answer to the previous conjecture. An important remark is that in profinite groups we will denote by w(G)the closure of the abstract subgroup generated by the set w{G}. In this setting, it is actually possible to prove that it is equivalent to formulate Conjecture 1.17 for profinite groups. Proposition 1.18. A word wis concise in all residually finite groups if and only if it is concise in all profinite groups. Proof. Let wbe a word that is concise in residually finite groups and suppose that w{G}is finite in a profinite group G. As Gis residually finite, the abstract subgroup generated by w{G}is finite too, but finite subsets are closed, and therefore w(G)is finite too. Suppose now that wis concise in profinite groups and assume that it takes finitely many values in a residually finite group G. Each residually finite groups embeds in its profinite completion b G. The first step is to prove that wtakes finitely many values in b G. Let g1, ..., gk∈b G. For each j= 1, . . . , k we can find a net of elements gj,i ∈G, indexed by a set I, such that limi∈Igj,i =gjand therefore w(g1, ..., gk) = limi∈Iw(g1,i, ..., gk,i)∈w{G}=w{G}, where the last equality is true because w{G}is finite hence closed. By hypothesis w(b G)≤b Gis finite and so w(G)is finite too. Any profinite group is either finite or uncountable. Detomi, Morigi and Shumyatsky realized that a similar duality could be valid also for word maps. For this reason they conjectured in [25] that any word taking countably many values in a profinite group has a finite verbal subgroup, proving the conjecture for outer commutators and other specific words. An improvement of this was obtained in [24], where the authors managed to avoid the dependence on the continuum hypothesis. 21 Definition 1.19 (Strongly concise words).A word wis said to be strongly concise if, whenever |w{G}| <2ℵ0in a profinite group G, then w(G)is finite. Detomi, Klopsch and Shumhyatsky proved that outer commutators and other specific words are indeed strongly concise, leading to a strengthening of Conjecture 1.17. Conjecture 1.20. Every word is strongly concise. In view of Theorem 1.15, we could ask whether words that are concise in residually finite groups are also boundedly concise in residually finite groups. This is currently unknown, because one essential tool in the proofs of Fernández-Alcober and Morigi or Mann in [31] was constructing an ultraproduct of groups. We cannot generalize their proof to residually finite groups because the ultraproduct of residually finite groups is not necessarily residually finite. For this reason, this is currently an open problem. Conjecture 1.21. Every word that is concise in residually finite groups is also boundedly concise in residually finite groups. 1.6 A comprehensive list of known concise words We will give a comprehensive list of all results regarding conciseness of words. As already mentioned, the first article that mentioned the problem is by TurnerSmith [80] in 1964, in which he proved that non-commutator words, lower central words and derived words are concise. Wilson proved that all outer commutator words are concise in [82] in 1974, but the proof is already more convoluted. It is important to mention that Fernández-Alcober and Morigi gave a different proof of this last result in [31]. This last proof developed new methods in the study of outer commutator words, by applying proofs by induction on the height and defect of these words, by representing them as finite trees. Apart from outer commutator words, the first type of words for which conciseness problems were extensively studied are Engel words. Indeed, in 2011 both Abdollahi and Russo [1] and Fernández-Alcober, Morigi and Traustason [32] proved that Engel words en= [x,ny]are concise for n≤4. These results rely heavily on the fact that any group in which e4is a law is locally nilpotent, whereas it is unknown if the same is true for the general n-Engel word en. The proof of Fernández-Alcober, Morigi and Traustason obtains some structural results for groups Gsuch that en{G}is finite for a certain positive integer n. Indeed, they proved that in this case [en(G), G]is a finite subgroup. 22 Another class of words that was studied are words obtained by nesting noncommutator words into outer commutator words. We will say that some words u1, . . . , unare disjoint if the sets of variables appearing in each of them are pairwise disjoint. In 2019 Delizia, Shumyatsky, Tortora and Tota proved in [22] that the word [u1, u2]is concise for u1, u2disjoint non-commutator words. This result was generalized by Azevedo and Shumyatsky in [9] to commutators [u1, u2, u3]for u1, u2, u3disjoint non-commutator words. In the same article, Azevedo and Shumuyatsky proved that, if u1, . . . , ukare disjoint copies of the same non-commutator word uand vis another non-commutator word disjoint from u1, . . . , uk, then both [u1, . . . , us]and [v, u1, . . . , us]are concise. Lastly, they proved that if uis an outer commutator word and vis a disjoint non-commutator word, then [u, v]is concise. In Chapter 2 we will give a full proof of a result that generalizes all of these. In [34], Fernández-Alcober and the author proved a stronger version of a conjecture of Azevedo and Shumyatsky, showing that, whenever u1, . . . , ukare non-commutator words, then the words γk(u1, . . . , uk)and δk(u1, . . . , u2k)are concise. Theorem 1.15 assures that any word that is concise is also boundedly concise. However, some results proved that some sets of words Ware uniformly boundedly concise, which means that for every w∈ W the same function fgives a bound as in Definition 1.14. In [13] Brazil, Krasilnikov and Shumyatsky proved that all lower central words and derived words are uniformly boundedly concise. This result was generalized to all outer commutator words by Fernández-Alcober and Morigi in [31]. Moving towards conciseness in some restricted classes of groups, we already mentioned that Turner-Smith proved that every word is concise in residually finite groups all whose quotients are residually finite. In 1967 an extremely important result of Merzljakov in [57] extended verbal conciseness to the class of groups such that, for each integer m∈N, there exists a finite index normal subgroup N(m)such that N(m)is residually (finite of order coprime to m). This result was used in Merzljakov’s article to prove that every finitely generated linear group is verbally concise. In this direction, a recent result of Zozaya [89] proved that the class of compact R-analytic groups is also verbally concise. In a similar way, there are other classes of groups that are verbally concise simply because no word can take finitely many values, like the class of groups that do not satisfy any law. This class of groups contains for example free groups and, as shown by Abért in [2], Thompson’s group F, weakly branch groups or profinite groups with alternating composition factors of unbounded degree. Conciseness for this class of groups follows from this easy lemma. 23 Lemma 1.22. If a word wtakes finitely many values in a group G, then Gsatisfies a law. Proof. Assume |w{G}| ≤ mand that wis a word in nvariables. Consider n× (m+ 1) variables, that we denote by xj i,i∈ {1, . . . , n},j∈ {1, . . . , m + 1}and define wa=w(xa 1, . . . , xa n),wa,b =w−1 awb. If γtis the simple commutator of length t=m(m−1)/2, the word γt(w1,2, . . . , w1,m+1, w2,3. . . , wm−1,m) obtained by computing γton all couples (a, b)∈ {1, . . . , m + 1}2with a < b is a law, because at least two of the wi,i∈ {1. . . , m + 1}must be equal. We will now discuss conciseness in residually finite and profinite groups. The first words that were proven to be concise in the class of residually finite groups, but which are not known to be concise in all groups, are words of the type wqfor wan outer commutator word and qa prime power. This was proved by Acciarri and Shumyatsky in [3], where they also showed that if wis a lower central word, then wqis boundedly concise in residually finite groups. In 2015 Guralnick and Shumyatsky proved that weakly rational words are concise in residually finite groups [38]. A word wis weakly rational if, for all finite groups Gand every integer ecoprime to |G|the set w{G}is closed by taking e-th powers. Burns and Medvedev in [14] defined that a word wimplies virtual nilpotency if every finitely generated metabelian group in which wis a law has a nilpotent subgroup of finite index. The authors proved that wimplies virtual nilpotency if and only if for all primes p,wis not a law in the wreath product CpoC∞. Some examples of words that imply virtual nilpotency are uv−1for u, v semigroup words in finitely many generators, Engel words and some generalizations of Engel words. In a series of two articles [26] and [27], Detomi, Morigi and Shumyatsky proved bounded conciseness in residually finite groups for words implying virtual nilpotency and several words of Engel type [w,ny]for npositive integer and wan outer commutator word. For w=γn kfor npositive integer they showed that both [w,ny]and [y,nw]are boundedly concise. If wis a prime power of an outer commutator word, they proved that [w,ny]is concise in residually finite groups, but it is unknown whether it is boundedly concise too. The best result in this direction was recently obtained by Acciarri and Shumyatsky in [4], showing that wand [w,ny]are concise in residually finite groups for wan arbitrary power of an outer commutator word and ya variable not appearing in w. 24 A more recent result of Azevedo and Shumyatsky [9] states that whenever u, v are two disjoint words, if uis concise in residually finite groups and vis a noncommutator word, then [u, v]is concise in residually finite groups. Moreover, if u is boundedly concise, then the same is true for [u, v]. In [25] Detomi, Morigi and Shumyatsky proved that if w{G}is countable in a profinite group Gfor w=x2,w= [x2, y]or wan outer commutator word, then w(G)is finite. All these results were generalized to the case |w{G}| <2ℵ0by Detomi, Klopsch and Shumyatsky in [24], where they obtained the same result also for the words w=x2,w=x3,w=x6,w= [x3, y],w= [x, y, y],w= [x, y, y, z1, . . . , zr],w= [x2, z1, . . . , zr]and w= [x3, z1, . . . , zr]where x, y, ziare different variables. In [48], Khukhro and Shumyatsky obtained strong conciseness for all Engel words w= [x,ny]in finitely generated profinite groups. We can also extend the notion of strong conciseness to some maps that are not word maps, like coprime and anti-coprime commutators. We will discuss these maps in detail in Chapter 4. We also mention some results on strong conciseness under the additional hypothesis that w(G)is generated by finitely many w-values. In [24] the authors proved that in this case, weakly rational words and words implying virtual nilpotency are strongly concise. Under the same hypothesis, Azevedo and Shumyatsky proved in [8] that [y,nvq]and [vq,n, y]are strongly concise for v=γk(x1, . . . , xk), and extended this result to some additional specific words under more conditions. Overall, Conjectures 1.17 and 1.20 are still widely open, but they have been partially settled for some specific subclasses of profinite groups. Indeed in [23] Detomi proved that every word is strongly concise in virtually nilpotent profinite groups, whereas in [4] Acciarri and Shumuyatsky proved that Conjecture 1.17 reduces to proving conciseness in the class of virtually pro-pgroups for an arbitrary prime p. 25 1.7 Tables of concise words We conclude the chapter with some tables summarizing the results we described, highlighting only the most general results. Concise words Words References Notes Non-commutators [81] (P. Hall) Outer commutators [82], [31] Uniformly concise [31] Engel words en,n≤4[1], [32] [en(G), G]is finite for every n[32] γk(u1, . . . , uk),δk(u1, . . . , u2k) uidisjoint non-commutators [34] Verbally concise classes of groups Class of groups References Res. finite with all quotients res. finite [81] Linear groups [57] Compact R-analytic [89] Groups without any law Lemma 1.22 We also remark that every word is strongly concise in virtually nilpotent profinite groups ([23]). 26 Words concise in residually finite groups Words References Notes wq,[wq,ny] wouter comm., q∈N[4] Weakly rational [38] Words implying virtual nilpotency [26] [wq,ny],[y,nwq] wouter comm., q∈N[27] boundedly concise for [γq k,ny],[y,nγq k] [u, v],u, v disjoint, uconcise in res. finite vnon-commutator [9] boundedly concise if uboundedly concise We will write (FG) for “w(G)is generated by finitely many w-values”. Strongly concise words Words References Notes Outer commutator [24] w=xq,q= 2,3,6 and some specific words [24] Engel words en,n∈N[48] For finitely generated profinite groups Coprime commutators γ∗ k,δ∗ k[39] Not group words see Chapter 4 Strongly concise words under additional conditions Weakly rational, implying virtual nilpotency [24] Condition (FG) [y,nγq k],[γq k,ny] y, γkdisjoint, q∈N and some specific words [8] Condition (FG) 27 28 2 Conciseness on normal subgroups In this chapter we describe some contributions to the list of known concise words. Delizia, Shumyatsky, Tortora, and Tota proved in [22] that, if u1and u2are noncommutator words in disjoint sets of variables, then [u1, u2]is concise too. This result has been extended to the case when u1is an outer commutator word and u2is a non-commutator and to commutators [u1, u2, u3]of non-commutators in [9]. For longer commutators, the only partial result was obtained by Azevedo and Shumyatsky in [9], who proved that if u1, . . . , ukare copies of the same noncommutator word in different variables, then [u1, . . . , uk]is concise. Azevedo and Shumyatsky conjectured that, if uiare non-commutator words in disjoint sets of variables and w=γk, then w(u1, . . . , uk)is concise. The aim of this chapter is to prove this conjecture, and moreover to extend it to the case of a generic outer commutator word w. We will roughly follow the article [34] of Fernández-Alcober and the author, where we proved these results for lower central words and derived words. In the first section we will develop some preliminary lemmas. These will be sufficient to settle the conjecture of Azevedo and Shumyatsky, for w=γk, in the second section. The main idea of the proof is to find a series of verbal subgroups such that each section of this series has some linearity properties. This could be obtained as a corollary of the case of generic outer commutator words, but the proof in this case is more straightforward and easier, so it makes sense to have a 29 1. Siis a normal subset of G. 2. There exists ni∈Nsuch that all nith powers of elements of Niare contained in Si. If for the tuple S= (S1, . . . , Sr)the set of values γr{S}is finite of order m, then the subgroup γr(N)is also finite, of (m, r, n1, . . . , nr)-bounded order. Proof. We follow the notation Niand Pr i, introduced in the statement of Theorem 2.11. We are going to prove that Pr iis finite of bounded order for i= 1, . . . , r + 1 by reverse induction on i. Since Pr 1=γr(N), this proves the result. The basis of the induction follows from Lemma 2.12, since we have that Pr r+1 = [γr(N), γr(N)]. Let us assume that Pr i+1 is finite of bounded order and prove that the same holds for Pr i. Recall that the quotient Pr i/Pr i+1 is the image of γr(Ni), and then, by a suitable application of Lemma 2.3, it can be generated by the images of the set Tof commutators [s1, . . . , si−1, xi, si+1, . . . , sr], with sj∈Sjfor 1≤j≤r,j6=i, and xi∈γi{Si}, where Si= (S1, . . . , Si). By Lemma 2.7, we have γi{Si} ⊆ S∗2i−1 i, and then Lemma 2.8 implies that [s1, . . . , si−1, xi, si+1, . . . , sr]∈γr{S}∗2i−1⊆γr{S}∗2r−1. From the assumption that |γr{S}| =m, we get |T| ≤ (2m+ 1)2r−1, and consequently Pr i/Pr i+1 can be generated by an (m, r)-bounded number of elements. Since Pr i/Pr i+1 is abelian, the proof will be complete once we show that all elements in Thave bounded finite order modulo Pr i+1. By Theorem 2.11, the word γris linear in position iof the tuple Nimodulo Pr i+1. In particular, [s1, . . . , si−1, xi, si+1, . . . , sr]λni≡[s1, . . . , si−1, xλni i, si+1, . . . , sr] (mod Pr i+1), (2.3) for every λ∈Z. Since xi∈γi(N1, . . . , Ni)≤Ni, it follows from (ii) in the statement of the theorem that xλni i∈Sifor all λ∈Z. Thus we get [s1, . . . , si−1, xλni i, si+1, . . . , sr]∈γr{S}. 36 Since γr{S}is finite of order m, it follows that there exist λ, µ ∈ {0, . . . , m}, λ6=µ, such that [s1, . . . , si−1, xi, si+1, . . . , sr]λni≡[s1, . . . , si−1, xi, si+1, . . . , sr]µni(mod Pr i+1). This implies that [s1, . . . , si−1, xi, si+1, . . . , sr]has (m, ni)-bounded finite order modulo Pr i+1, as desired. If we take Si=Ni, we get Theorem 2.2 for the lower central words. Corollary 2.14. Let r∈Nand let N= (N1, . . . , Nr)be a tuple of normal subgroups of a group G. If γr{N}is finite of order m, then the subgroup γr(N)is also finite, of (m, r)-bounded order. Now we deduce Theorem 2.1 for lower central words. Corollary 2.15. Let r∈Nand let u1, . . . , urbe disjoint non-commutator words. Then the word γr(u1, . . . , ur)is boundedly concise. In particular, γr(xn1 1, . . . , xnr r) is boundedly concise for all ni∈Z r {0}. Proof. Let us consider the word w=γr(u1, . . . , ur), and let Gbe a group in which wtakes finitely many values, say |w{G}| =m. By Corollary 2.5, we have w(G) = γr(u1(G), . . . , ur(G)). Note that ui(G) = hSii, where Si=ui{G}, and that w{G}=γr{S}, where S= (S1, . . . , Sr). Now observe that Siis a normal subset of Gand that, since uiis a non-commutator word, for some ni∈Z r {0} we have {gni|g∈G} ⊆ ui{G}. Hence w(G)is finite of (m, r, n1, . . . , nr)-bounded order by Theorem 2.13. 2.3 An example: the word δ2 We now want to prove Theorems the analogues of Theorems 2.1 and 2.2 for a generic outer commutator word w. The general strategy is still the same as for lower central words: we are going to obtain a suitable series of normal subgroups of G, going from [w(N), w(N)] to w(N), with the property that each of the factors of the series can be generated by a verbal subgroup on a tuple of normal subgroups that is closely related to w(N)and linear in one component. This is basically Theorem 2.20 below. For simplicity, let us refer to such a series as a linear series. The argument needed to obtain a linear series for derived words presents difficulties and subtleties that did not arise with lower central words, and is also significantly 37 more technical. For the convenience of the reader and in order to make the procedure for a general wmore understandable, first of all we are going to provide a sketch of it in the particular case of δ2. Of course, δ1=γ2and, according to Theorem 2.11, we have the following linear series for δ1(N1, N2): [N1, N2] N1,[N1, N2] [N1, N2],[N1, N2] Figure 2.1: Series of [N1, N2] In this and in the next diagrams, a red box indicates the component in which we have linearity. Let us see how we can construct a linear series for δ2(N1, N2, N3, N4)from the series above for δ1. To this purpose, we will use Lemma 2.10, which ensures that linearity is preserved after taking suitable commutators, and also the remark made before that lemma, saying that linearity is preserved after multiplying by a normal subgroup. To start with, we take the commutator of the terms of the previous series with [N3, N4], obtaining the series [N1, N2],[N3, N4] hN1,[N1, N2],[N3, N4]i h[N1, N2],[N1, N2],[N3, N4]i 38 Now we multiply this series by [N1, N2],[[N1, N2],[N3, N4]], which contains the subgroup [N1, N2],[N1, N2],[N3, N4]by P. Hall’s Three Subgroup Lemma, and we get the following diagram: [N1, N2],[N3, N4] hN1,[N1, N2],[N3, N4]i h[N1, N2],[N1, N2],[N3, N4]i Figure 2.2: First diagram for [N1, N2],[N3, N4] Here, and in the remaining diagrams, instead of the subgroups of the series, we are showing verbal subgroups on normal subgroups whose images coincide with the corresponding factors of the series. After all, it is in these subgroups where we are going to obtain the linearity conditions. Be aware then that vertical lines in the diagrams do not denote inclusions from this point onwards. By swapping the roles of (N1, N2)and (N3, N4), we can obtain this other diagram: [N1, N2],[N3, N4] h[N1, N2],N3,[N3, N4]i h[N1, N2],[N3, N4],[N3, N4]i Figure 2.3: Second diagram for [N1, N2],[N3, N4] Now we take the commutator of [N1, N2]with the terms of this last diagram, and we add the extra term δ2(N1, N2, N3, N4)′at the bottom: 39 h[N1, N2],[N1, N2],[N3, N4]i h[N1, N2],[N1, N2],N3,[N3, N4]i h[N1, N2],[N1, N2],[N3, N4],[N3, N4]i h[N1, N2],[N3, N4],[N1, N2],[N3, N4]i Figure 2.4: Series of h[N1, N2],[N1, N2],[N3, N4]i Finally, by gluing the diagrams in Figures 2 and 4 together, we obtain a linear series for the subgroup δ2(N1, N2, N3, N4). Of course, this is simply a sketch without proofs, but we are going to follow the same procedure in the proof of Theorem 2.20, in order to get a linear series for w= [α, β]from the series for the outer commutator words αand β. At this point, it is worth noting an important difference with the situation for a lower central word γr. In that case, every factor of the linear series is of the following form (again we show the linear component in red): N1, . . . , Ni−1,[N1, . . . , Ni], Ni+1, . . . , Nr. We observe that this subgroup is of the form γr(M), where the jth component Mj of Mis either Njor a commutator of the terms of Nthat involves Nj, and the linearity happens in Mi. However, if we look at the series for δ2obtained above, the first two subgroups in Figure 2.4 are δ2(N1;N2; [N1, N2]; [ N3, N4]) (2.4) and δ2(N1;N2; [N1, N2]; [N3,[N3, N4]]),(2.5) which are not of the form δ2(M)with every Mja commutator from Ninvolving Nj, as we can see by looking at the third component of δ2. Also, the linearity does 40 not happen in a component of δ2, but in a more interior position. Nevertheless, we can write these subgroups as verbal subgroups on normal subgroups for outer commutator words different from δ2. More specifically, if v(x1, x2, x3, x4, y1, y2) = [x1, x2],[[y1, y2],[x3, x4]] then the subgroups in (2.4) and (2.5) are v(M1)and v(M2), where M1= (N1, N2, N3, N4, N1, N2)and M2= (N1, N2, N3,[N3, N4], N1, N2),(2.6) where again we have marked the linear components in red. 2.4 Outer commutator words After having illustrated the procedure with the case of δ2, let us proceed to systematically develop the tools that are necessary for the proof of Theorem 2.20. We start by introducing a special type of words that we can derive from a given outer commutator word w, which we call extended words of w. Before giving the definition, we show the idea behind extended words with an example. Consider the word δ2= [[x1, x2],[x3, x4]]. This is formed by taking the commutator of x1 and x2, taking the commutator of x3and x4, and then taking the commutator of these two commutators. Now suppose that on some occasions, before performing one of these commutators, we introduce a change by taking first the commutator of one (or both) of the components with an outer commutator word not involving the variables x1, . . . , x4appearing in δ2. For example, before producing [x1, x2], we take the commutator [[y1, y2], x1]and now we follow as in δ2taking the commutator with x2, obtaining [[y1, y2], x1], x2. We could continue with the process of taking commutators without making any other changes, so getting [[y1, y2], x1], x2,[x3, x4], but we could also make some similar changes in the process, as in the words h[[y1, y2], x1], x2,x3,[y3, x4]i and h[[y1, y2], x1], x2,x3,[y3, x4], y4i. Another possibility is to make a commutator at the very end, after having completed δ2, as in y1,[[x1, x2],[x3, x4]]. 41 Observe that all these extended words are again outer commutator words, because we never repeat a variable when we make changes in the construction of δ2. Let us now give the formal definition of extended words. Notice that this definition differs from the one of extensions of outer commutator words given in Definition 3.1 of [33]. Definition 2.16 (Extended words).Let w=w(x1, . . . , xr)be an outer commutator, and let Y={yn}n∈Nbe a set of variables that are disjoint from X. For every k∈N∪ {0}, we define recursively the set extk(w)of kth extended words of was follows: 1. ext0(w) = {w}. 2. For k≥1,extk(w)consists of the set {[p, q],[q, p]|pouter commutator in Y,q∈extk−1(w),pand qdisjoint} ={[p, q]|pouter commutator in Y,q∈extk−1(w),pand qdisjoint}±1, and, if w= [α, β], also of the set S ℓ+m=k {[p, q]|p∈extℓ(α),q∈extm(β),pand qdisjoint}. If v∈extk(w)then we say that wis an extended word of degree kof wby outer commutators. For brevity, in the remainder we will simply speak of extended words when we mean extended words by outer commutators. Observe that an extended word vof an outer commutator w=w(x1, . . . , xr)is again an outer commutator, in the variables {x1, . . . , xr} ∪ Y. Whenever it is convenient we will assume, after renaming the variables, that v=v(x1, . . . , xr, yr+1, . . . , ys). Next we generalize Lemma 2.8 to extended words of an outer commutator word. Lemma 2.17. Let v=v(x1, . . . , xr, yr+1, . . . , ys)be an extended word of degree k of an outer commutator word w=w(x1, . . . , xr). Assume that S= (S1, . . . , Sr)is a tuple of normal subsets of a group G. If t= (t1, . . . , ts)is a tuple of elements of Gsuch that ti∈S∗mi ifor every i= 1, . . . , r, then v(t)∈w{S}∗m1...mr2k. 42 Proof. We use induction on k+r. If k= 0 then v=wand the result is Lemma 2.8. This gives in particular the basis of the induction. Suppose now that the result holds for smaller values of k+r, and that k≥1. According to Definition 2.16, we may assume that v(t) = [p(t′), q(t′′)], where pand qare disjoint and 1. either pis an outer commutator word in Yand q∈extk−1(w), 2. or p∈extℓ(α),q∈extm(β), with w= [α, β]and ℓ+m=k. In case (i), all elements t1, . . . , trappear in the vector t′′, and by the induction hypothesis we have q(t′′)∈w{S}∗m1...mr2k−1. Then the result follows by applying Lemma 2.7 to the commutator word [x1, x2]and the normal subset w{S}∗m1...mr2k−1. Suppose now that we are in case (ii), and assume without loss of generality that α=α(x1, . . . , xq)and β=β(xq+1, . . . , xr). Set S′= (S1, . . . , Sq)and S′′ = (Sq+1, . . . , Sr). Since αand βinvolve less variables than w, the result is true for pand q, and so p(t′)∈α{S′}∗m1...mq2ℓand q(t′′)∈β{S′′}∗mq+1...mr2m. Now the result follows by applying Lemma 2.8 to the commutator word [x1, x2] and the pair of normal subsets (α{S′}, β{S′′}). We also need to define a type of extensions of tuples of normal subgroups and of verbal subgroups on normal subgroups. The idea behind the definition is to be able to deal with tuples like the ones appearing in (2.6) and with the corresponding verbal subgroups on normal subgroups in that paragraph. Definition 2.18 (Outer commutator extension).Let Gbe a group and consider two tuples N= (N1, . . . , Nr)and M= (M1, . . . , Ms)of normal subgroups of G. We say that Mis an outer commutator extension of Nif the following conditions hold: 1. s≥r. 2. For every i= 1, . . . , s, we have Mi=wi(Ni), where wiis an outer commutator word and all components of Nibelong to N. 3. For every i= 1, . . . , r, the subgroup Niis a component of Ni, and consequently Mi≤Ni. 43 Definition 2.19 (Extensions of w(N)).Let w=w(x1, . . . , xr)be a word and let Nbe an r-tuple of normal subgroups of a group G. An extension of degree kof w(N)by outer commutators is a subgroup of the form v(M), where vis an extended word of degree kof wand Mis an outer commutator extension of N. For example, we can see the subgroup in (2.5) as an extension of δ2(N) = δ2(N1, N2, N3, N4)by taking v=[x1, x2],[[y1, y2],[x3, x4]]and the tuple M= (N1, N2, N3,[N3, N4], N1, N2). Note that v(M)is linear in the fourth component modulo the subgroup that appears below it in Figure 4. We now prove the existence of a linear series for outer commutator words. We recall that the height of an outer commutator word w= [α, β]is defined inductively, with a single variable having height 0, and with the height of wbeing 1 + max{height(α),height(β)}. Notice that the height of an outer commutator word in svariables will always be at most s−1. Theorem 2.20. Let r∈Nand let N= (N1, . . . , Nr)be a tuple of normal subgroups of a group G. Consider an outer commutator word w= [α, β]in r variables, say of height h. Then there exists a series [w(N), w(N)] = V0≤V1≤ · · · ≤ Vt=w(N) of normal subgroups of Gsuch that, for every i= 1, . . . , t, the following hold: 1. The section Vi/Vi−1is the image of an extension vi(Mi)of w(N)of degree at most h−1. 2. In the section Vi/Vi−1, the word viis linear in one component of the tuple Mi. Furthermore, the words viand the words appearing in the outer commutator extensions Midepend only on wand r, and not on the group Gor on the tuple N. Proof. We prove the theorem by induction on the height of the outer commutator word w, with the base case being a single variable, which is obvious. We can then assume that there exist two series of subgroups satisfying the conditions of the theorem for the outer commutator words of smaller height αand β. Assume that x1, . . . , xqand xq+1, . . . , xq+mare the variables involved in αand βrespectively, and in particular r=q+m. Set N1= (N1, . . . , Nq)and N2= (Nq+1, . . . , Nq+m). By the induction hypothesis, there exist two series of length sand rrespectively A0= [α(N1), α(N1)] ≤ · · · ≤ Ai≤ · · · ≤ As=α(N1)(2.7) 44 and B0= [β(N2), β(N2)] ≤ · · · ≤ Bi≤ · · · ≤ Br=β(N2)(2.8) such that, for every i= 1, . . . , s, the factors Ai/Ai−1and Bi/Bi−1are the images of vα i(Mα i)and vβ i(Mβ i), respectively, where: (a) vα iand vβ iare extended words of αand βrespectively, each of degree at most h−2. (b) Mα iis an outer commutator extension of N1. (c) Mβ iis an outer commutator extension of N2. (d) In the sections Ai/Ai−1and Bi/Bi−1, the words vα iand vβ iare linear in one component of the tuples Mα iand Mβ i, respectively. Let us now see how to obtain the series for wand for the tuple Nfrom the two series (2.7) and (2.8). We will have that the length tof the series we are looking for depends on the length of these two series, in the form that t=r+s+ 1. We start by taking the commutator of all terms of the series (2.7) with β(N2). This way we obtain the series [A0, β(N2)] ≤ · · · ≤ [Ai, β(N2)] ≤ · · · ≤ [α(N1), β(N2)] = w(N).(2.9) By P. Hall’s Three Subgroup Lemma, we have [A0, β(N2)] = [α(N1), α(N1), β(N2)] ≤[α(N1), β(N2), α(N1)] = [α(N1), w(N)]. Now we multiply all terms of the series (2.9) by [α(N1), w(N)], and this is the rightmost part of the series we are seeking (where t=r+s+ 1, as above): Vt−s= [α(N1), w(N)] ≤ · · · ≤ Vt−s+i= [Ai, β(N2)] [α(N1), w(N)] ≤ · · · ≤ Vt=w(N).(2.10) Note that t−s=r+ 1. The factors in this series are the images of the subgroups [vα i(Mα i), β(N2)], which can be represented in the form vi(Mi)by taking vi= [vα i, β(xq+1, . . . , xq+m)] 45 3.1 Elements of small cancellation theory In this section we introduce some basics in small cancellation theory, which are crucial for the construction of the counterexamples to Hall’s conjecture. We start with some notation. Given a surface or a polygon X, we denote by ∂(X)the boundary of Xand by ι(X) = Xr∂(X)the interior of X. If we view an edge Xof a polygon as a polygon itself, then ∂(X)consists of the two endpoints. When defining diagrams over groups, if a group Ghas a set Sof generators, it will be useful to consider the set S∗of abstract words in the alphabet S∪S−1. In accordance to the notation introduced by Olshanskii, in this chapter we will denote words in S∗by capital letters C, L, M, X, Y, Z. We will write |X|to denote the length of the word X∈S∗and, if X, Y ∈S∗, we will write X≡Y(and say that Xand Yare visually equal) if |X|=|Y|and we have a letter-by-letter equality. Definition 3.1 (Cells).Consider a n-gon Pin a plane with edges e1, . . . , en. Consider a map f:P→X, where Xis any surface, such that: •f|ι(P)is an embedding; •f|ι(ei)is an embedding for each i∈ {1, . . . , n}; • if a, b ∈Pare distinct points with f(a) = f(b), then a, b ∈∂(P). If ais a vertex, so is b, otherwise if a∈ei,b∈ej, then f(ei) = f(ej). The image f(P)of such a map is called a cell on X. Informally, a cell is an identification of the n-agon Pin X, but we allow vertices and edges to be pasted together by f, still preserving the structure of open disc of ι(P). Definition 3.2 (Cell decomposition).Acell decomposition of a surface Xis a finite set {(Pi, fi)|i= 1, . . . , m}of cells such that X=Sm i=1 fi(Pi)and such that fi(Pi)∩fj(Pj),i6=jis either empty or it is a set of vertices and/or edges. A cell decomposition can be thought as a partition of Xinto a finite set of cells, but allowing these cells to intersect in edges and/or vertices. The images of vertices or edges of any of the Piwill be called vertices and edges of the cell decomposition. Normally, we will denote a cell fi(Pi)with a single letter C. Even if the theory can be developed for arbitrary surfaces, we will only work with orientable surfaces. It will be useful to give an orientation to edges of a 52 cell decomposition, by assuming that any edge eadmits an inverse e−1, which geometrically corresponds to the same element, but with inverse orientation. Fix now an alphabet Sand assign to each oriented edge eof the cell decomposition a label ϕ(e)∈S∪S−1. If these labels are chosen such that ϕ(e−1) = ϕ(e)−1 for each edge eof the cell decomposition, we will moreover say that the decomposition is a diagram. If pis a path, obtained by concatenating some oriented edges p=e1· · · ekthe label of pis the word ϕ(p) = ϕ(e1)· · · ϕ(ek)∈S∗, where the endpoint of eicoincides with the beginning of ei+1. Whenever the surface Xunderlying a diagram is a disc, it will be called a circular diagram. Notice that if Xhas a boundary, then it must consist of edges and vertices of the diagram, and therefore each of its connected components will have a label as a path. We will say that any connected component of the boundary ∂X of Xis a contour of X. Moreover any cell Cof a diagram can be seen as a disc (possibly with some parts of the boundary pasted together), and in this case the contour is equal to the boundary, and thus we will use the same notation ∂C. We will use the convention that the label of the contour of every cell of a diagram over an orientable surface will be read clockwise, and similarly for the label of the contour of a circular diagram. If we have a cell Cin a diagram, the boundary ∂Cis a path (induced by the orientation of the polygons), so we can always consider the label ϕ(∂C)of the contour. The length of the contour of a cell or of a surface Xis the number of edges of ∂X as a finite path and will be denoted by |∂X|. Normally, we simply study groups given by a presentation G=hS| Ri. In our case, we will need to consider groups with graded relations, in the sense that we partition the set Rof relations as R=S∞ i=1 Riin such a way that no relator in Rican coincide with a cyclic conjugate of a word in Rjor its inverse if j6=i. In this setting, we will consider the graded presentation G=S| R = ∞ [ i=1 Ri.(3.1) A cell in a diagram ∆is an R-cell if its label is visually equal (up to cyclic conjugation) to a relator or an inverse of a relator in R. If such relator is in the set Ri, we will say that the cell is an i-cell, or a cell of rank i. Moreover we will say that it is a 0-cell (or a cell of rank 0) if its label is visually equal to a word ss−1or s−1sfor s∈S. Definition 3.3 (Diagram over a group).If Gis a group given by the presentation (3.1), a diagram over Gis a diagram ∆over the alphabet Ssuch that all cells 53 are either R-cells or 0-cells. The rank of the diagram is the maximum among the ranks of its cells. Notice that this definition depends on the presentation (3.1) chosen for G, not only on the group Gitself. When studying diagrams over a group G, we want to study the simplest possible version of them. In our case, we will say that a circular subdiagram of rank jcan be simplified if we can substitute it with a subdiagram of smaller rank with the same countour label. As it is shown in Section 13.2 of [66], a sequence of these operations can always lead to a reduced diagram, that is a diagram without subgraphs that can be simplified. In 1933 van Kampen proved that some fundamental problems in group theory, like understanding if a word in the generators is the trivial element in the group, can be solved through the use of diagrams of groups. We will give a version of his result for reduced diagrams over graded groups, and refer to Theorem 13.1 of [66] for the proof. Theorem 3.4 (van Kampen).Let Wbe a non-empty word in the alphabet S∪S−1. Then W= 1 in a group Gwith graded presentation (3.1) if and only if there exists a reduced circular diagram over Gsuch that the label of its contour is visually equal to W. w r1 r2 Figure 3.1: Van Kampen’s Lemma The name “small cancellation theory” is due to the fact that we often require that different relators have a small overlapping. This can be made more precise with the following definition. Definition 3.5 (Pieces and Condition C′(λ)).Let Gbe a group with graded presentation (3.1), and let R1, R2,R16=R2, be two cyclic conjugates of two relators or inverses of relators in R. A word Xin the alphabet S∪S−1is a piece if R1and R2are visually equal to words of the form XY1and XY2respectively. 54 The presentation (3.1) satisfies small cancellation condition C′(λ)for a number 0< λ ≤1if, whenever Ris a cyclic conjugate of a relator, or of an inverse of a relator, such that it is visually equal to XY for a piece X, then |X|< λ|R|. Example 3.6. The group ha, b |aba−1b−1isatisfies C′(λ)for all λ > 1/4. Pieces consist of single letters or their inverses. The surface group ha, b, c, d |[a, b][c, d]]isatisfies C′(λ)for all λ > 1/8, and as before pieces consist of single letters or their inverses. It is possible to see that any group can have a presentation satisfying condition C′(λ)for λ > 1 5(Gol’berg, see Section 12.4 of [66]), but if we ask λto be smaller, it allows us to obtain interesting conditions on the groups. Theorem 3.7 (Greendlinger’s, Theorem 12.1 [66]).Let ∆be a reduced circular diagram over a presentation of a group Gthat satisfies C′(λ)for λ≤1 6with at least one R-cell. Suppose that the label ϕ(∂∆) is cyclically reduced and has no proper subword equal to the identity. Then there exists an exterior arc p(i.e. a path p∈∂C ∩∂∆) of some R-cell Csatisfying |p|>1 2|∂C|. This theorem has crucial implications in combinatorial group theory because, if there exists a presentation (3.1) of a group Gsatisfying C′(λ)for λ≤1 6, it is possible to define an algorithm (called Dehn’s algorithm) that in a finite number of steps can recognize if a word W∈S∗is equal to the identity in G. Moreover this combinatorial condition has strong geometric implications, namely the group Gis an hyperbolic group. Even if the presentations we will use will not satisfy any C′(λ)condition, we will find an an analogue of Greendlinger’s Theorem in groups satisfying weaker small cancellation conditions. 3.2 Ivanov’s counterexample In 1989 Ivanov obtained the first counterexample to P.Hall’s conjecture stating that all words are concise in the class of all groups. More precisely, he provided a word wIand a group Iin which wIis not concise. Theorem 3.8 ([43]).The word wI(x, y) = [[xpn, ypn]n, ypn]n for n > 1010 odd and p > 5000 prime, takes only two values {1, z}in a torsionfree 2-generated group Ibut the verbal subgroup wI(I) = hzi, that corresponds to the center of the group I, is infinite cyclic. 55 This group is a central extension of an infinite two generated group GI(∞)of bounded exponent, which is constructed using small cancellation theory. We first need a crucial result in central extensions. We recall that a set Rof relations for a group G=hS| Ri is independent if no proper set R′⊆ R of relations gives the same group G(with the identity map on S). Theorem 3.9. Suppose that the group G=hS| Ri can be considered as G∼ = F/N, with Fbeing the free group with basis Sand N=hRi. Then •G=F/[F, N]is a central extension of G=F/N, i.e. N=N[F, N]is contained in the center of G. Moreover, if Gis centerless, then N=Z(G); •G=hS|[r, s]for r∈ R, s ∈Si; •if Ris an independent set of relations for G, then Nis a free abelian group with basis R=R[F, N]. For the proof, we refer to Chapter 31 of [66], in particular to Theorem 31.1 and the discussion above. In the following we construct the group I, giving an idea of the arguments involved in the proof that wItakes a single non-trivial value in I. In order to do so, we first construct the group GI(∞), which is a variation of the free Burnside group constructed by Olshanskii in [64] by inductively imposing (possibly different) torsion to elements of a free group, and by obtaining a torsion group as the limit of all of these quotients. Let F2=F(a, b)the free group in two letters and let V, W ∈F2be elements of the free group. Denote by |V|the minimal length of Vas a word in the alphabet {a, b, a−1, b−1}. Fix an ordering in F2such that if |V|<|W|, then V < W (but we do not necessarily have to choose lexicographic order for words of the same length). For each i≥1we inductively construct the groups GI(i). Define GI(0) = F2, then assume we already constructed GI(i−1). Let Ci∈F2be the smallest word (with respect to the fixed ordering of F2) corresponding to an element of infinite order in GI(i−1), such a word will be called period of rank i. Define GI(i) = GI(i−1)/hCni iiGI(i−1), where hCni iiGI(i−1) is the normal closure of Cni iand niis a odd number greater than n= 1010. The limit of these quotients is the group GI(∞) = F2/hCni i|i∈NiF2.(3.2) When ni=nfor every iwe obtain the free Burnside group B(2, n). In this construction, by using diagrams on groups, Olshanskii proved that the set {Cni i|i∈ 56 N}is an independent set of relations (i.e. no proper subsets of relations defines the same group GI(∞)), that every word of finite order in GI(i)is conjugate to a power of a period Cjfor j≤i(so in the torsion group GI(∞)all words are conjugate to a power of a period), and most importantly that the group obtained in this way is infinite and with trivial center. Ivanov’s group Iwill be obtained as a central extension of GI(∞)for some specific choices of the exponents ni. In this case, we must choose some relators in a different way and we will impose some specific periods to have different orders. In detail, for n > 1010 odd and p > 5000 prime we require that the words Ci satisfy that: • the smallest word of length 1 is C1=B1=aand we impose it to have order p2n; • the smallest word Ciof length 4(pn+1) is B2= [bapnb−1, apn]and we impose it to have order pn; • the 8 smallest words Ciof length 8n(pn + 1) will be B3, . . . , B10 [[bε1aε2pnb−ε1, aε3pn]n, aε3pn] for ε1, ε2, ε3∈ {±1}. We impose these 8 words to have order n; • all the other words Ciwill have order n. In Lemma 2 of [43] it is proved that the word Cihas infinite order in GI(i−1) (and hence it can be chosen to be a period of appropriate rank) and that the group GI(∞)with presentation (3.2) obtained by imposing these restrictions is infinite. The group Iis obtained as the quotient I=F2/h[Cni i, T], Cni i=Cnj j|i, j ∈N, T ∈F2iF2. In accordance to Theorem 3.9, if we considered only the first set of relators {[Cni i, T]}we would obtain a central extension of GI(∞). The center of such a group would be a free abelian group with infinite basis {Cni i|n∈N}, but by adding the relators Cni i=Cnj jfor all i, j ∈N, we obtain a cyclic center, generated by a single element z=Cni ifor every i∈N. Now we want to show that the only non-trivial value taken by the word wIin the group Iis exactly z. Following the steps of a proof in [64], Ivanov proved the following result (Lemma 1 of [43]), which has a clear analogy to Theorem 3.7. 57 Lemma 3.10. Let ∆be a reduced annular diagram or diagram on a disc with two holes over GI(∞)(with presentation 3.2, Cias in the previous paragraph) such that the labels of the contour segments are cyclically unshortenable. If ∆contains at least an R-cell, then there exists a cell Cwith ∂C ∩∂∆ = pfor a path psuch that |p| ≥ 10−4|∂C|. To show that zis indeed the only value assumed by the word, we need to “funnel” the values of some subwords of the word wI. We will explicitly explain the first steps to show the use of diagrams over groups and to understand why we need the diversification of the exponents. Suppose first that Xand Yare two words in the alphabet S={a, b}. We first assume that [Xpn, Y pn]n= 1 in GI(∞), in which case [Xpn, Y pn]nis in the center of Iso v(X, Y ) = 1. As all words are conjugate to a power of a period in GI(∞), if Xor Yare conjugate to a period different than C1=a, then either Xpn or Ypn are equal to the identity in GI(∞)(as all the periods different from C1=ahave order dividing pn) and we are in the case [Xpn, Y pn]n= 1 in GI(∞), that we already considered. We can therefore assume that X=L−1at1Land Y=M−1at2Mfor some words L, M ∈S∗,t1, t2∈Z, and that [Xpn, Y pn]n6= 1 in GI(∞). In this case, the word [Xpn, Y pn]must be conjugate in GI(∞)to a power of B1=aor B2, being the only periods with order not dividing nin GI(∞). We want to prove that it cannot be conjugate to a power of a. Suppose by contradiction it is possible. Then, for a certain N∈S∗, we would have [L−1at1pnL, M−1at2pnM] = N−1at3N. By Theorem 3.4, and then pasting the paths of the contour with label Nand N−1, we can construct a diagram ∆over GI(∞)on a disc with an hole such that the exterior contour has label at3and the interior contour, read clockwise, has label [L−1at1pnL, M−1at2pnM](see Figure 3.2). We can now paste together the two segments of the internal contour with label LM−1at2pnML−1and its inverse respectively, so we get a diagram on a disc with two holes and the contours are at3,at1pn and a−t1pn (Figure 3.3). By refining it if necessary, we can assume the diagram to be reduced. 58 at3 [L−1at1pnL, M−1at2pnM] N Figure 3.2: Pasting contours with label N at3 a−t1pn LM−1at2pnML−1 at1pn Figure 3.3: Pasting contours with label LM−1at2pnML−1 Now use small cancellation theory: by Lemma 3.10, if ∆contains at least an R-cell, there exists a cell Cwith contour label Cnj jfor a certain j∈Nsuch that it has a boundary arc pof length at least 10−4|Cnj j|. As 10−4ni≥2for every i∈N, C2 jmust be a subword of akfor k=±t1pn or k=±t3, so Cj=C1=a. We can now excise the cell C, in the sense that we remove Cand, if ∂C=pq with pbeing the boundary arc C ∩ δ∆, the new contour of ∆will follow the path qin place of the previous boundary arc p(Figure 3.4). 59 ak1ak2ak2 ak4 ak3ak3 ap2nExcision Figure 3.4: Excision of a cell By excising all the cells of this type, with labels a±p2n, we change the exponents of some labels of the contour, but not their residual class modulo p2n. After having excised all these cells, we have a diagram on a disc with two holes and by Lemma 3.10 it cannot contain any R-cell (and in particular the disc with two holes is degenerate, with no interior, Figure 3.5). Looking at the final diagram, we can notice that the label of the exterior boundary is equal to the label of the path obtained by concatenating the two interior boundaries. This implies that t3≡pn(t1−t1)≡0 (mod p2n)so [L−1at1pnL, M−1at2pnM] = N−1at3N= 1 in GI(∞), and this contradicts our assumptions. ak1ak2 ak3 Figure 3.5: Final diagram, after exicisions With similar arguments, by means of congruences preserved by cell excision in diagrams, Ivanov proved that if [Xpn, Y pn]n6= 1, then this word is conjugate to either B2or B−1 2, not to a proper power of them. Still applying the same ideas, but with more complicate congruences, he also proved that [[Xpn, Y pn]n, Y pn]must be a conjugate of exactly one of the words B3, . . . , B10. We refer to the last part of 60 Lemma 3 in [43] for the explicit computations. This is sufficient to conclude that the only non-identical value of the word must be z. A further interesting remark is that, as it is written in the acknowledgements of [43], the anonymous referee claimed that, using Adian’s arguments of [6], the word wA= [xr, yr]ntakes exactly two values in a central extension (with cyclic center) of the free Burnside group B(2, n)for odd n= 3r≥1005. This claim has not been proved, but it would provide the first example of a word that is concise (in this case [xr, yr], see [22], or Theorem 2.13) but such that its power is not concise. Notice that if such a claim was true, using that each inverse of a wA-value is still a wA-value, the word wAwould have to take at least three values (the identity, a non-trivial element of infinite order and its inverse). Even with this correction, this remains only a claim and the proof is not a straightforward adaptation of Ivanov’s methods. In view of Conjectures 1.17 and 1.20, it is natural to ask whether the counterexample obtained by Ivanov provides a negative answer to these questions too. However, it is well known that the group Iis not residually finite, thus cannot be used to obtain a counterexample to the aforementioned conjectures in a straightforward way. We now provide a proof of this fact. Lemma 3.11. Let Gbe a residually finite group and let Nbe the marginal subgroup of a word w(x1, . . . , xn). Then G/N is residually finite. Proof. Let g∈Gsuch that g /∈N, in particular there exists an index i∈ {1, . . . , n}and some elements h1, . . . , hn∈Gsuch that t=w(h1, . . . , hig,...,hn)w(h1, . . . , hi, . . . , hn)−16= 1 By residually finiteness there exists a normal subgroup Mof finite index in G such that t /∈M. We claim that g /∈MN. If it was, let m∈M, n ∈Nsuch that g=mn. As Nis marginal for w, we would have that t=w(h1, . . . , himn, . . . , hn)w(h1, . . . , hi, . . . , hn)−1= w(h1, . . . , him, . . . , hn)w(h1, . . . , hi, . . . , hn)−1 but this would imply that t≡1 (mod M), contradicting our choice of M. This proves that for every g /∈Nthere exists a finite index subgroup MN such that g /∈MN, as desired. Corollary 3.12. Any finitely generated group which is central-by-(infinite group of finite exponent) cannot be residually finite. In particular, Ivanov’s group Iis not residually finite. 61 3.5 On generation of verbal subgroups In [24], the authors proved strong conciseness of several classes of group words, like words implying virtual nilpotency or weakly rational words, under the additional assumption that, if the verbal subgroup w(G)is finitely generated, then it can be generated by finitely many w-values. If w(G)is a pro-pgroup, then by looking at the quotient w(G)/Φ(w(G)) and using Burnside basis theorem, it is immediate to see that w(G)is finitely generated if and only if it is generated by finitely many w-values. The authors asked whether this is always true. Conjecture 3.21. Let Gbe a profinite group and wbe a word. If w(G)is topologically finitely generated, it can be generated by finitely many word values. We will now show that this question has a negative answer for lower central words w=γk. Theorem 3.22. There is a profinite group Gsuch that the subgroup γk(G)is procyclic for every k, but it cannot be generated by finitely many γk-values. Clearly the group Gin our construction cannot be finitely generated otherwise, as Nikolov and Segal proved in [59][60], all the abstract subgroups of the lower central series would be closed. In that case, whenever a verbal subgroup w(G)is finitely generated, it is also generated by finitely many w-values: since it coincides with an abstract subgroup that is finitely generated, each generator is a finite word in the alphabet w{G}, so the subgroup itself is also generated by finitely many w-values. A special case of the question we are interested in is when the derived subgroup is procyclic. Under this more restrictive hypothesis, is it true that it is generated by a single commutator? This question was studied, in the setting of abstract groups and cyclic subgroups, by Macdonald in [56]. He proved the following result. Theorem 3.23 (Macdonald).Let Gbe an abstract group and assume G′is cyclic. If either Gis nilpotent or G′is infinite, then G′is generated by a suitable commutator. In general, for any given positive integer k, there is a finite group Mksuch that M′ kis cyclic but it cannot be generated by less than kcommutators. The main tool in our proof is the group Mkin the second part of the proposition, so we will give an idea of its structure. Fixed k, set m= 22k−1and pick a set of different odd primes p1, . . . , pm, chosen arbitrarily. The group Mkwill be a semidirect product CoC2k 2, where C2k 2=ha1, . . . , a2kiis the direct product of 68 2kcopies of the cyclic group of order 2 and C=hciis a cyclic group of order p1p2· · · pm. Assume [c, ai] = cαifor some integer αi. Macdonald proved, with the use of some accurately chosen congruences, that it is possible to select the integers αiin the construction of Mkin such a way that the derived subgroup is the whole Cand that for any set of k−1commutators g1, . . . , gk−1there is a prime pj∈ {p1, . . . , pm} such that hg1, . . . , gk−1i=hcpji. In [45], Kappe observed that γ2{Mk}=γj{Mk}for all j≥2, hence the following result is a direct consequence of Macdonald’s theorem. Corollary 3.24 (Kappe).For any given positive integer kand any j≥2, there is a finite group Mksuch that γj(Mk)is cyclic but it cannot be generated by less than kcommutators. Proof of Theorem 3.22. We will prove the result for the commutator subgroup. The same construction works for all lower central words by Corollary 3.24. For every k, let Mkbe the group constructed by Macdonald in Theorem 3.23. In the choices of the group Mk, we had to choose some odd primes pk 1, . . . , pk 22k−1, we can require all of them to be different both pairwise and from all primes pj lfor 1< j < k and 1< l < 22j−1. Define Bi=Qi k=1 Mk.By construction, B′ iis a direct product of cyclic subgroups of coprime order, so it is cyclic too. As M′ icannot be generated by less than icommutators, the same is true for B′ i. Moreover, by construction the groups Bi, for i∈N, form an inverse system of finite groups, so we can define the profinite group G= lim ←− Bi. Clearly G′is procyclic as it is the inverse limit of B′ i. It cannot be generated by a finite set Xof commutators, say of cardinality n, because otherwise the images of Xin the quotient Bn+1 would generate B′ n+1 too, contradicting the previous paragraph. In the way the authors use this property in [24], it is still relevant to ask whether the same phenomena can happen under the assumption that |w{G}| <2ℵ0. Of course, in view of Conjecture 1.20, it is possible that no word can take countably many values in a group unless it takes finitely many, so this could be considered as an intermediate step towards the proof of the strong conciseness conjecture. Conjecture 3.25. Let Gbe a profinite group and wbe a word. If w(G)is topologically finitely generated and |w{G}| <2ℵ0, then w(G)can be generated by finitely many word values. 69 Clearly the example in Theorem 3.22 cannot be used in order to contradict this conjecture because lower central words are strongly concise (see [24]). 70 4 Coprime commutators In this chapter we prove strong conciseness of coprime commutators γ∗ kand δ∗ k, following the article [39] by de las Heras, Shumyatsky and the author. We will first define coprime commutators and give an overview of their history. Coprime commutators are not word maps, but behave in a similar way, and they constitute a good generating set of the pronilpotent residual (for γ∗ k,k≥2) or of the k-th pronilpotent residual (for δ∗ k,k≥1). In the second section we will discuss some basic lemmas that are necessary to develop our results. Some of them were already present in the literature and others are original. In the third section we will outline the structure of the proofs, with a description of an interesting set of pronilpotent subgroups that is present in prosolvable groups. We will then prove the main theorems, of strong conciseness of coprime commutators, both in the meta-pronilpotent case for γ∗ kin Section 4, and in the (prosolvable of Fitting heigth k+ 1) case for δ∗ kin Section 5. The general statements will be then proved jointly in Section 6. 71 4.1 History of coprime commutators Higher order coprime commutators were introduced by Pavel Shumyatsky in [75] as a way to obtain a smaller natural set of generators for some classical subgroups. Given a profinite group Gand an element x∈G, we denote by |G|(respectively |x|) the order of G(respectively x) as a supernatural number and π(G)(respectively π(x)) will stand for the set of prime numbers dividing |G|(respectively |x|). We will say that an element g∈Gis a simple coprime commutator if and only if it can be written as g= [g1, g2]for g1, g2∈Gwith (|g1|,|g2|) = 1. It was already well-known that the set of simple coprime commutators in a finite group Ggenerates the nilpotent residual γ∞(G), that is, the smallest normal subgroup Nsuch that G/N is nilpotent (see Theorem 2.1 of [75]). Of course, in profinite groups the pronilpotent residual γ∞(G) = Tiγi(G)is the intersection of the terms of the lower central series of G. Coprime commutators of higher order were defined in [75] for finite groups, but the definition naturally extends to the profinite case. Definition 4.1 (Higher order coprime commutators).Let γ∗ 1{G}=δ∗ 0{G}=G and, for every positive integer idefine inductively the sets γ∗ i{G}=[xλ, g]|x∈γ∗ i−1{G}, λ ∈b Z, g ∈G, (|xλ|,|g|) = 1 δ∗ i{G}=[xλ1, yλ2]|x, y ∈δ∗ i−1{G}, λ1, λ2∈b Z,(|xλ1|,|yλ2|) = 1. Moreover, for the generated subgroups we will write γ∗ i(G) = hγ∗ i{G}i and δ∗ i(G) = hδ∗ i{G}i. Even if coprime commutators are not word maps, the analogy with classical word maps is clear. Indeed, γ∗ i(G)and δ∗ i(G)are fully invariant subgroups because the order of f(x)always divides the order of xfor every homomorphism fand every x∈G. For this reason, it is interesting to describe the subgroups generated by coprime commutators, and the main results of the article [75] completely solve this natural question. Theorem 4.2 ([75] Theorems 2.1, 2.7).Let Gbe a profinite group. If k≥2, the subgroup γ∗ k(G)is trivial if and only if Gis pronilpotent. The subgroup δ∗ k(G)is trivial if and only if Gis prosolvable of Fitting height at most k. 72 It is interesting to point out that a consequence of Theorem 4.2 is that there exists no word w∈F(X∞)such that w(G) = γ∗ i(G)(i= 2,3, . . .) or w(G) = δ∗ i(G)(ipositive integer) for every profinite group Gbecause nilpotent groups of unbounded class do not form a variety of groups. Several problems, that were classical for usual commutators, were then adapted to coprime commutators. An example is Ore’s Conjecture, which stated that every element of a finite simple group is a commutator, and was solved in [53]. In [75], the author conjectured that every element of a finite simple group can be realized as a coprime commutator and proved the conjecture for the class of alternating groups. The same conjecture was later settled for PSL2(q)for every prime power qin [67] and for Suzuki groups 2B2(q)for every odd qin [88]. Another natural consequence of the analogy between coprime commutators and usual commutators was the study of conciseness problems for them. Of course we will say that γ∗ i(resp δ∗ i) is concise if γ∗ i(G)(resp. δ∗ i(G)) is finite whenever γ∗ i{G} (resp. δ∗ i{G}) is finite. In [5] the authors proved that, if there exists a positive integer msuch that the word γ∗ ior δ∗ itakes at most mvalues in a finite group G, then the generated subgroup has m-bounded order. The bound does not depend on i, so that coprime commutators are uniformly concise in the class of finite groups. A straightforward consequence is that coprime commutators of higher order are concise in residually finite groups. In the article [24], that began the investigation in strong conciseness, the authors noticed that the concept of strong conciseness can be applied in a wider context. Suppose Cis a class of profinite groups and ϕ{G}is a subset of Gfor every G∈ C. Is the subgroup generated by ϕ{G}finite whenever |ϕ{G}| <2ℵ0? Such map ϕis said to be strongly concise in the class Cif the answer is positive. This question is interesting whenever ϕ{G}is defined in some natural way and/or properties of the subgroup hϕ{G}i have strong impact on the structure of G. For this reason, in [28] the authors examined strong conciseness for coprime commutators and managed to set that the map γ∗ 2is strongly concise in profinite groups. In this chapter, which roughly follows the article [39], we will prove strong conciseness of γ∗ iand δ∗ ifor every positive integer i. Theorem 4.3. A profinite group Gis finite-by-pronilpotent if and only if there is ksuch that the set of γ∗ k-values in Ghas cardinality smaller than 2ℵ0. Theorem 4.4. A profinite group Gis finite-by-(prosolvable of Fitting height at most k) if and only if the set of δ∗ k-values in Ghas cardinality smaller than 2ℵ0. 73 Of course there are results of strong conciseness because by Theorem 4.2 the values of the words γ∗ kand δ∗ kgenerate the finite subgroups of Theorems 4.3 and 4.4. 4.2 Preliminaries We will first list some results that were present in the literature, or some small variations of them, that will be useful in the proofs of Theorems 4.3 and 4.4. The first one is a fundamental result in the study of strong conciseness. A direct application of this result is that conjugacy classes in profinite groups are either finite or of cardinality at least 2ℵ0(see Lemma 3.17). Proposition 4.5 ([24] Lemma 2.1).Let φ:X→Ybe a continuous map between two non-empty profinite spaces that is nowhere locally constant (i.e. there is no non-empty open subset U⊆oXwhere φ|Uis constant). Then |φ(X)| ≥ 2ℵ0. A classical result in the theory of coprime automorphisms is the following. Lemma 4.6 ([42], Lemma 4.29).Let Abe a group of automorphisms of a finite group Gwith (|A|,|G|) = 1. Then, [G, A] = [G, A, A]. The following lemma is a stronger version of this result for the case where Gis a pronilpotent group. Lemma 4.7 ([47] Lemma 4.6).Let φbe an automorphism of a pronilpotent group Gwith (|φ|,|G|)=1. Define the set the set S={[g, φ]|g∈G}. Then the map θ:S→Sdefined as θ:x→[x, φ] is bijective. The following is a profinite version of Lemma 2.4 in [75]. Lemma 4.8. Let Gbe a profinite group and let g1, . . . , gkbe δ∗ k−1-values in G. Suppose that g1, . . . , gk∈NG(H)for a subgroup H≤Gwith (|H|,|gi|) = 1 for every i∈ {1, . . . , k}. Then, for every h∈H, the element [h, g1, . . . , gk]is a δ∗ k-value. Using the previous two lemmas together, we will be able to guarantee that some special types of long commutators are also values of δ∗ k. 74 Lemma 4.9. Let G1, . . . , Gkbe pronilpotent subgroups of a profinite group G such that Gj≤NG(Gi)for all j≤i. Let xi∈Gifor every iand assume that (|xi|,|xi+1|)=1for all i= 1, . . . , k. Then the element g= [x1, . . . , xk]is in δ∗ k−1{G}and π(g)⊆π(xk). Proof. We will prove by induction on ithat gi:= [x1, . . . , xi]∈δ∗ i−1{G}for every i∈ {1, . . . , k}and that π(gi)⊆π(xi). The statement of the lemma corresponds to the case i=k. If i= 1 the result is obvious, so assume i > 1and that gi−1is aδ∗ i−2-value with π(gi−1)⊆π(xi−1), so in particular (|gi−1|,|xi|) = 1. If His the minimal Hall subgroup of the pronilpotent group Gicontaining xi, then gi−1acts as a coprime automorphism of H. By Lemma 4.7, there exists yi∈Hsuch that [xi, gi−1] = [yi, gi−1,i−1 . . ., gi−1], and Lemma 4.8 shows that gi= [xi, gi−1]is a δ∗ i−1-value, as desired. As gi∈H, we immediately have that π(gi)⊆π(xi). The next result is a profinite version of Lemma 2.4 in [5]. We recall that by “meta-pronilpotent” group we mean a profinite group Ghaving a normal pronilpotent subgroup Nsuch that G/N is pronilpotent. Lemma 4.10. Let Gbe a meta-pronilpotent group. Then γ∞(G) = Qp[Kp, Hp′], where Kpis a Sylow p-subgroup of γ∞(G)and Hp′is a Hall p′-subgroup of G. For a general group word w, the set w{G}of w-values of a profinite group G is always closed in G. We will show that the same is true for the sets of γ∗ kand δ∗ k-values. Proposition 4.11. Let S1, . . . , Skbe closed subsets of a profinite group G. Then the set C={(g1, . . . , gk)∈S1× · · · × Sk|(|gi|,|gi+1|) = 1 for all i= 1, . . . , k −1} is closed in S1× · · · × Sk. Furthermore, the sets γ∗ k{G}and δ∗ k{G}are closed in G. Proof. Let Pbe the set of all primes and p∈ P. First notice that for every closed subset Sof Gthe set Sp′={g∈S|p /∈π(g)} is closed. Indeed Sp′=TN⊴oGSp′Nbecause p∈π(g)if and only if there is a normal subgroup Nsuch that gN has order divided by pin G/N. Also, the set 75 S b Z={gλ|g∈S, λ ∈b Z}is the image under the continuous map f(g, λ) = gλof the compact set S×b Z, so it is closed too. Let now A, B be subsets of G. We claim that the set RA,B =\ p∈P (A×Bp′)∪(Ap′×B)(4.1) is exactly the set of elements (a, b)∈A×Bwith |a|and |b|coprime. On the one hand, if |a|and |b|are coprime then (a, b)∈(A×Bp′)∪(Ap′×B)for every p∈ P, because, if b∈BrBp′, then a∈Ap′necessarily. On the other hand, if (a, b)∈RA,B and a prime pdivides |a|, then (a, b)∈A×Bp′so pdoes not divide |b|, and the claim follows. Notice now that if Aand Bare closed, the set RA,B is an intersection of closed subsets of G×Gso it is closed too. It is now easy to prove by induction on kthat the sets γ∗ k{G},δ∗ k{G}are closed: just note that γ∗ k{G}is exactly the set RA,B in (4.1) with A= (γ∗ k−1{G}) b Z,B=G, whereas δ∗ k{G}is the set RA,B in (4.1) with A=B= (δ∗ k−1{G}) b Z. To prove that the set Cis closed in S1× · · · × Sk, it suffices to notice that by the above arguments the set Ci=S1× · · · × Si−1×RSi,Si+1 ×Si+2 × · · · × Sk is closed for every i∈ {1, . . . , k −1}and C=Tk−1 i=1 Ci. As we showed in Lemma 1.12, whenever a group word wtakes finitely many values in a group G, the subgroup w(G)is finite if and only if w(G)/w(G)′is finite. If wtakes less than 2ℵ0values in Gwe cannot obtain the same conclusion in general, but with some slightly stronger hypothesis we can anyway obtain a similar result. Lemma 4.12. Let ϕbe a map that associates to every group Ga normal subset ϕ{G} ⊆ G. Let Gbe a profinite group with |ϕ{G}| <2ℵ0and let Kbe a pronilpotent subgroup of hϕ{G}i generated by a subset of ϕ(G). If K/K′is finite, then Kis finite. Proof. Since Kis pronilpotent, we have K′≤Φ(K), where Φ(K)stands for the Frattini subgroup of K. Thus K/Φ(K)is finite, and hence we can find a finite subset Sof ϕ{G}generating K. Since ϕ(G)is a normal subset of G, by Lemma 3.17 each of these generators has finitely many conjugates in G, so in particular |G:CG(s)|<∞for every s∈S. Since CG(K) = Ts∈SCG(s), this implies that Z(K) = K∩CG(K)has finite index in K, and by Schur’s theorem K′is finite. 76 We will use Lemma 4.12 for ϕ=γ∗ kor ϕ=δ∗ k, but it could be applied to other cases, such as any group word map or uniform (anti-coprime) commutators (see [28] or [29]). 4.3 Introduction to the proofs In order to fully understand the proofs of Theorems 4.3 and 4.4, we have to begin from the proof of Detomi, Morigi and Shumyatsky in [28] that settled the analogous result for γ∗ 2. In the aforementioned article, the authors first proved that γ∗ 2is strongly concise in meta-pronilpotent groups and then used this partial result to settle the general case. We will similarly split our proof: first we will prove strong conciseness of γ∗ k in meta-pronilpotent groups (Proposition 4.20 in Section 4.4), then we will settle the problem for δ∗ kin prosolvable groups of Fitting height k+ 1 (Proposition 4.33 in Section 4.5) and we will use these partial results in the proof of Theorems 4.3 and 4.4, that will be proved jointly in Section 4.6. The proof of Proposition 4.20 consists of extending the reasoning that was used in [28] for γ∗ 2, with a focal use of Lemma 4.7. The proof of the general case also partially follows [28], with some complications in the arguments. The case of δ∗ kin prosolvable groups of Fitting height k+ 1, however, involved a lot of technical problems and is surely the more complex part of this chapter. For this reason, in this case we give a deeper analysis and motivation of the ideas involved. An essential tool of the proof is the following collection of subgroups. Definition 4.13 (Sylow basis).ASylow basis of a profinite group Gis a family {Pi}of Sylow subgroups of G, one for each prime in π(G), such that PiPj=PjPi for every i, j. The normalizer of a Sylow basis is T=TiNG(Pi). Basic properties of Sylow bases for finite groups can be found in Section 9.2 of [73] and they extend naturally to profinite groups. Lemma 4.14. Any prosolvable group admits a Sulow basis and any two Sylow bases are conjugate. In this case, the Sylow basis normalizer Tis pronilpotent and G=Tγ∞(G). Moreover, if Gis meta-pronilpotent, γ∞(G) = [T, γ∞(G)]. Proof. The first statement is a classical result, see for example Proposition 2.3.9 of [72], whereas the fact that G=Tγ∞(G)is Lemma 5.6 of [69]. If Gis metapronilpotent, we have that γ∞(G) = [G, γ∞(G)] = [Tγ∞(G), γ∞(G)] = [T, γ∞(G)] 77 Proof. Assume first that ℓ6= 1, and proceed by induction on t−ℓ. If t−ℓ= 0, then [g1, . . . , gt−1, g′ tgt] = [g1, . . . , g′ t]gt[g1,...,gt]−1[g1, . . . , gt], and the result follows. Assume t−ℓ > 0, and we write, for the sake of brevity, y= [g1, . . . , gℓ−1]. By induction, we have [y, g′ ℓgℓ, gℓ+1, . . . , gt] = [[y, g′ ℓ, ghℓ ℓ+1, . . . , ght−2 t−1]ht−1[g1, . . . , gt−1], gt] with hi∈Gℓ· · · Gifor i∈ {ℓ, . . . , t −1}. Now, [[y, g′ ℓ, ghℓ ℓ+1, . . . , ght−2 t−1]ht−1[g1, . . . , gt−1], gt] = [[y, g′ ℓ, ghℓ ℓ+1, . . . , ght−2 t−1]ht−1, gt][g1,...,gt−1][g1, . . . , gt] = [y, g′ ℓ, ghℓ ℓ+1, . . . , ght−2 t−1, g(ht−1)−1 t]ht−1[g1,...,gt−1][g1, . . . , gt], and the lemma follows. If ℓ= 1, a similar argument applies. Lemma 4.22. Let G1, . . . , Gtbe subgroups of a profinite group Gsuch that Gj≤NG(Gi)for every j≤i. Let ℓ∈ {1, . . . , t}and Y1, Y2⊆Gℓbe such that π(y1), π(y2)⊆π(y1y2)for every y1∈Y1,y2∈Y2. Let Xi⊆Gifor i∈ {1, . . . , ℓ −1}, and for i∈ {ℓ+ 1, . . . , t}denote Xi=Gi. Then: 1. If φ∗ {ℓ}(Y1;Xi) = φ∗ {ℓ}(Y2;Xi) = 1, then φ∗ {ℓ}(Y1Y2;Xi) = 1. 2. If φ∗ {ℓ}(Yj;Xi) = ∅for some j∈ {1,2}, then φ∗ {ℓ}(Y1Y2;Xi) = ∅. Proof. Since π(y1), π(y2)⊆π(y1y2)for every y1∈Y1,y2∈Y2, the second statement is straightforward. Moreover, if φ{ℓ}(y1y2;gi)∈φ∗ {ℓ}(Y1Y2;Xi), then for j∈ {1,2}we have φ{ℓ}(yj;gi)∈φ∗ {ℓ}(Yj;Xi). The result follows now directly from Lemma 4.21. In view of the preceding lemma, we now introduce a convenient way to choose coset representatives of normal subgroups. These will play an important role throughout the chapter. Definition 4.23 (Good representatives).Let Gbe a profinite group and U⊴G. An element g∈Gis a good representative of the coset gU if π(g), π(u)⊆π(gu) for every u∈U. Lemma 4.24. Let Ube an open normal subgroup of a pronilpotent group G. Let gbe a representative of the coset gU and write g=Qp∈π(G)gpwith gpap-element of G. Then the following are equivalent: 84 (i) gis a good representative of the coset gU; (ii) gp= 1 whenever gp∈Ufor p∈π(G); (iii) π(g)is minimal among all representatives of the coset gU. In this case, if σ=π(G/U), then π(g)⊆σ. Proof. We first prove (i)⇒(ii). Assume gis a good representative and suppose that gp∈U. If gp6= 1, then π(g·g−1 p)does not contain p, contradicting that π(g)⊆π(gu)for all u∈U. (ii)⇒(i). Write u=Qp∈π(G)upfor a certain u∈Uand suppose gp= 1 whenever gp∈U. Then, if either gp6= 1 or up6= 1, then gpup6= 1, that is exactly the condition of being a good representative. (ii)⇔(iii) is immediate, and the last remark follows from (ii). The following lemma is an application of Proposition 4.5 to a special type of coprime commutators. Lemma 4.25. Let G1, . . . , Gtbe pronilpotent subgroups of a profinite group Gsuch that Gj≤NG(Gi)for all j≤i, and |δ∗ t−1{G}| <2ℵ0. For every i∈ {1, . . . , t}, let Sibe a closed subset of Gi. If φ∗(Si)6=∅, then, there exist elements xi∈Giand open subgroups Ui⊴oGisuch that |φ∗(xiUi∩Si)|= 1. Proof. Let C=n(x1, . . . , xt)∈S1× · · · × St(|xi|,|xi+1|) = 1 for all i= 1, . . . , to. As φ(C) = φ∗(Si), we have C 6=∅. Note that Cis closed in G1× · · · × Gtby Lemma 4.11. Fix (x1, . . . , xt)∈ C. By Lemma 4.9 the element gk:= [x1, . . . , xt]is in δ∗ t−1{G}. Hence, |Imm(φ)|<2ℵ0, and by Proposition 4.5, it follows that there exist elements xi∈Giand open normal subgroups Ui⊴Gisuch that C ∩ (x1U1× · · · × xtUt)6=∅ and |φ∗(xiUi∩Si)|= 1. Lemma 4.25 will often provide some cosets of open subgroups of Gin which coprime commutators are trivial. Lemmas 4.26 and 4.29 below will allow us to relate coprime commutators of these cosets with coprime commutators of the open subgroups themselves. 85 Lemma 4.26. Let G1, . . . , Gtbe subgroups of a profinite group Gsuch that Gj≤ NG(Gi)for every j≤i, and for every i∈ {1, . . . , t}, let xi∈Giand Ui⊴Gi. Assume also that Gj≤NG(Ui)for every j≤i. Fix j∈ {1, . . . , t}and write J={1, . . . , j −1}, then: (i) If φ(xiUi) = 1 then φJ(xiUi;Ui) = 1. (ii) If φJ(xiUi;Ui) = 1 then φJ∪{j}(xiUi;Ui) = φ(x1U1, . . . , xj−1Uj−1, xj, Uj+1, . . . , Ut). Proof. (i) We will proceed by reverse induction on j∈ {1, . . . , t + 1}, where the base case j=t+ 1 translates to φ(xiUi)=1, which is true by hypothesis. Let thus j < t + 1 and assume that φJ∪{j}(xiUi;Ui) = 1. Let Ct= 1 and for every i∈ {j+1, . . . , t−1}define Ci=CUi(Ui+1/Ci+1). Note that Ciis well-defined, since using that for every ℓthe subgroup Uℓis normal in G1· · · Gℓ, one can easily show by induction that Cℓ⊴G1· · · Gℓ. If j≥2, let Y={[x1u1, . . . , xj−1uj−1]|ui∈Ui, i = 1, . . . , j −1}. Then, we can rewrite φJ∪{j}(xiUi;Ui) = 1 as [Y, xjUj]⊆CGj(Uj+1/Cj+1). For every i∈ {1, . . . , j}, fix ui∈Uiand shorten y= [x1u1, . . . , xj−1uj−1]. Then we have [y, xjuj] = [y, uj][y, xj]uj, and since CGj(Uj+1/Cj+1)is a normal subgroup of Gjcontaining [y, xjuj]and [y, xj], it follows that [y, uj]∈CGj(Uj+1/Cj+1). This shows that φ(x1U1, . . . , xj−1Uj−1, Uj, Uj+1, . . . , Ut) = 1, as we wanted. For the case j= 1, note that both x1and x1U1lay in CG1(U2/C2), so that U1≤CG1(U2/C2). (ii) For every i∈ {j+ 1, . . . , t}we define Cias in (i). For i∈ {1, . . . , t}, let ui∈Uiand shorten y= [x1u1, . . . , xj−1uj−1]. Then, [y, xjuj] = [y, u′xj] = [y, xj][y, u′]xj= [y, u′]xi[xj,y][y, xj] for some u′∈Uj, and note that z:= [y, u′]xj[xj,y]∈CGj(Uj+1/Cj+1). Then [z, u′ j+1, . . . , u′ t] = 1 for every u′ i∈Ui,i∈ {j+ 1, . . . , t}, so that [y, xjuj, uj+1, . . . , ut] = [z[y, xj], uj+1, . . . , ut] = [y, xj, uj+1, . . . , ut], where the last equality follows from Lemma 4.21. The lemma follows. 86 Definition 4.27 (Subgroup Nσ).Let G1, . . . , Gtbe pronilpotent subgroups of a profinite group Gsuch that Gj≤NG(Gi)for all j≤i. Let σbe a finite set of primes. We define the normal subgroup Nσ=hφ∗ {j}(Hi;Gi)|jis such that |π(Gj)|=∞iG, where Hiis the Hall σ-subgroup of Gifor every i. If |π(Gi)|<∞for all i, then Nσ=h∅iG= 1 for every σ. The subgroups G1, . . . , Gtof Gfor which the definition of Nσapplies will be clear from the context. Notice that for any finite sets of primes σ1and σ2such that σ1⊆σ2we have Nσ1≤Nσ2.(4.4) Lemma 4.28. Let G1, . . . , Gtbe pronilpotent subgroups of a profinite group G such that Gj≤NG(Gi)for all j≤i. Fix ℓ∈ {1, . . . , t}and xℓ∈Gℓ. For i∈ {1, . . . , ℓ −1}, let Xi⊆Gi, and for i∈ {ℓ, . . . , t}let Ui⊴oGibe such that Gj≤NG(Ui)for j≤i. Suppose that (|xℓ|,|xℓ−1|)=(|xℓ|,|Uℓ+1|)=1for every xℓ−1∈Xℓ−1. If φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , Ut) = 1, then we have φ∗(X1, . . . , Xℓ−1, Uℓ, . . . , Ut) = 1. Proof. First of all, observe that since φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , Ut)6=∅, there are y1, . . . , yℓ−1such that yi∈Xiand (|yj|,|yj+1|) = 1 (4.5) for all j∈ {1, . . . , ℓ −2}. Note that the tuple (y1, . . . , yℓ−1,1, . . . , 1) is in Cand then φ∗(X1, . . . , Xℓ−1, Uℓ, . . . , Ut)6=∅. Fix then a tuple (x1, . . . , xℓ−1, uℓ, . . . , ut)∈ C with xj∈Xjand uj∈Uj. In order to conclude we want to prove that φ∗(x1, . . . , xℓ−1, uℓ, . . . , ut) = 1. For i∈ {ℓ, . . . , t}, let Hibe the minimal Hall subgroup of Uicontaining ui, and notice that we have (|xℓ−1|,|Hℓ|) = (|Hj|,|Hj+1|) = 1 (4.6) for all j∈ {ℓ, . . . , t −1}. Since Gℓis pronilpotent, we have π(xℓh)⊆π(xℓ)∪ π(h)for all h∈Hℓ, and hence, as (|xℓ|,|xℓ−1|) = (|xℓ|,|Uℓ+1|) = 1, we have φ(x1, . . . , xℓ−1, xℓHℓ, Hℓ+1, . . . , Ht)⊆φ∗(X1, . . . , Xℓ−1, xℓUℓ, Uℓ+1, . . . , Ut), and it is then equal to the trivial subgroup. Lemma 4.26(i) now gives φ(x1, . . . , xℓ−1, Hℓ, . . . , Ht) = 1, and therefore we have φ∗(x1, . . . , xℓ−1, uℓ, . . . , ut) = 1. Lemma 4.29. Let Gi, ℓ, Xi, Uibe as in Lemma 4.28. 87 (i) For i∈ {ℓ, . . . , t}, suppose that either |π(Gi)|=∞, in which case we write Yi=Gi, or |π(Gi)|= 1, in which case we write Yi=Ui. Assume moreover that if π(Gi) = {p}consists of a single prime, then p /∈π(Gi−1)∪π(Gi+1). Suppose we also have that φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , xtUt) = 1 for some xℓ∈Gℓsuch that (|xℓ|,|xℓ−1|)=1for every xℓ−1∈Xℓ−1. Then, there exists a finite set of primes σsuch that φ∗(X1, . . . , Xℓ−1, Yℓ, . . . , Yt)⊆Nσ (cf. Definition 4.27). (ii) Suppose that we fix xi∈Gi,i=ℓ, . . . , t, such that (|xi|,|xi+1|) = 1 for all i∈ {ℓ, . . . , t −1}and (|xℓ|,|xℓ−1|) = 1 for all xℓ−1∈Xℓ−1. If the set φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , xtUt)is empty, then we also have that φ∗(X1, . . . , Xℓ−1, Gℓ, . . . , Gt) = ∅. Proof. (i) Write L={ℓ, . . . , t}, and for i∈L, define σi=       π(Gi/Ui)if |π(Gi)|=∞, π(Gi)if |π(Gi)|= 1. Let σ=σℓ∪ · · · ∪ σt. Up to changing the representative, we can assume that every xiis a good representative of xiUi, and in particular that they are all σ-elements by Lemma 4.24. Furthermore, since φ∗ L(xiUi;Xi)6=∅and π(xj)⊆π(xjuj)for every uj∈Uj, it follows that (|xi|,|xi+1|) = 1 for all i∈ {ℓ, . . . , t −1}. For i∈Lwith |π(Gi)|=∞, let Vibe the Hall σ′-subgroup of Gi, and for i∈Lwith |π(G)|= 1, set Vi=Ui(notice that Vi≤Uiif |π(Gi)|=∞). We want to apply Lemma 4.28 t−ℓ+ 1 times, first to the index t, then decreasing until we reach the index ℓ, with the Vitaking the role of the Ui. Say we are applying it to the index ℓ≤j≤tand let us check that the two coprimality conditions of Lemma 4.28 are satisfied. We first check the hypothesis (|xj|,|Vj+1|)=1. If |π(Gj+1)|=∞, then π(Vj)⊆σ′and the hypothesis is satisfied. If π(Gj+1) = {p}, then p /∈π(Gj)and in particular p /∈π(xj). As for the other condition, if j=ℓ, it is simply one of the hypotheses of the lemma. If ℓ+ 1 ≤j≤t, we have that (|xj|,|xj−1|) = 1 and (|xj|,|vj−1|) = 1 for all vj−1∈Vj−1, either because Vj−1is a σ′-subgroup if |π(Gj−1)|=∞or by hypothesis if |π(Gj−1)|= 1. At the end of this process we obtain φ∗ L(Vi;Xi) = 1. Now, if |π(Gi)|= 1, then Yi=Ui=Vi. If |π(Gi)|=∞, writing Hjfor the Hall σ-subgroup of 88 Gj, then φ∗(X1, . . . , Xℓ, Gℓ+1, . . . , Gj−1, Hj, Gj+1, . . . , Gt)⊆Nσby definition and by Lemma 4.22(i) we obtain that φ∗ L(Yi;Xi)⊆Nσ. (ii) If φ∗(X1, . . . , Xℓ−1, xℓUℓ, . . . , xtUt) = ∅then in particular we have that φ∗(X1, . . . , Xℓ−1, xℓ, . . . , xt) = ∅. The only way for this to happen is that there exists an index j∈ {1, . . . , l −2}such that (|xj|,|xj+1|)6= 1 for all xj∈Xj,xj+1 ∈Xj+1, and the lemma follows. The following lemma is the focal point of the proof of Proposition 4.33, as it will allow us to funnel some values of certain coprime commutators into an accurately chosen subgroup. Lemma 4.30. Let G1, . . . , Gtbe pronilpotent subgroups of a profinite group G such that Gj≤NG(Gi)for all j≤i, and |δ∗ t−1{G}| <2ℵ0. Then, there exist a finite set W⊆φ∗(Gi)and a finite set σof primes such that φ∗(Gi)⊆NσhWiG. As this is the most technical proof, we will first give an example of the procedure for a specific case to clarify the main ideas. Example 4.31. We restrict to the case t= 2, so we are studying φ∗(G1, G2), in the specific case when |π(G1)|= 1,|π(G2)|=∞and π(G1)∩π(G2) = ∅. Notice that for t= 2 some easier reasoning could lead to an analogous result, but we will follow the algorithm beneath the proof of Lemma 4.30 in order to illustrate it. By Lemma 4.25, for i∈ {1,2}, we obtain Ui⊴oGiand xi∈Gisuch that φ∗(x1U1, x2U2) = {w}consists of a single value. Set W={w}, we will work in G/hWiGand assume w= 1. We recall that by Remark 4.16, we can always refine an open normal subgroup U2⊴G2with another normal open subgroup which is normalized by G1too, so we will always assume that G1≤NG(U2). Lemma 4.29 (with ℓ= 1) gives a set σ(∅)of primes such that φ∗(U1, G2)⊆ Nσ(∅). We can factor out this subgroup and assume φ∗(U1, G2) = 1. Fix now a set S={s1= 1, . . . , sm}of coset representatives of U1in G1. As 1∈Sand π(G1) = 1, every element of Sis a good representative for U1. Set now V0=G2. For every ℓ∈ {1, . . . , m}, if φ∗(sℓ, Vℓ−1) = ∅, then set Vℓ=Vℓ−1, otherwise Lemma 4.25 gives a coset Vℓ⊆Vℓ−1, such that φ∗(sℓ, Vℓ) = 1. Notice that each Vℓis a coset of an open subgroup of G2. Repeating this procedure mtimes we get Vm=gV for V⊴oG2,g∈G2such that φ∗(sℓ, gV )is either empty or consists of the trivial element for every ℓ= 1, . . . , m. Notice that, being 1∈S, the set φ∗(S, gV )is non-empty. 89 Applying now Lemma 4.29, this time with ℓ= 2, we can obtain a finite set of primes σsatisfying φ∗(S, G2)⊆Nσ. Now, if we work in G/Nσ, we can apply Lemma 4.22 and obtain that φ∗(sℓU1, G2)is either empty or trivial for every ℓ∈ {1, . . . , m}. As Swas a set of coset representatives of U1in G1, we have that φ∗(G1, G2) = 1. Since the beginning of the proof, we have factored out the normal subgroups hWiGand Nσ(∅)∪σ, settling Lemma 4.30 in our case. Overall with several subgroups G1, . . . Gtsome additional steps might be necessary, but this case exemplifies the main ideas of the proof. Proof of Lemma 4.30. Let I={i∈ {1, . . . , t} | |π(Gi)|=∞}. It suffices to prove the theorem in the case when |π(Gi)|= 1 for all Giwith i /∈ I. The general case, where each Gi,i /∈ I, is the product of its Sylow subgroups follows by applying Lemma 4.22. For i /∈ I, let pibe a prime such that π(Gi) = {pi}. Then we have φ∗(Gi) = φ∗(G1, . . . , Gi−2, Hi−1, Gi, Hi+1, Gi+2, . . . , Gt), where Hi−1and Hi+1 are the Hall p′ i-subgroups of Gi−1and Gi+1, respectively. We can therefore assume, again by Lemma 4.22(i), that for all i /∈ I we have pi/∈π(Gi−1)∪π(Gi+1).(4.7) We claim that that for every J⊆ {1, . . . , t}rIthere exist a finite set WJ⊆ φ∗(Gi), a finite set of primes σ(J)and subgroups UJ i⊴oGiwith i /∈ I ∪ Jsuch that φ∗ I∪J(Gi;UJ i)⊆Nσ(J)hWJiG. We proceed by induction on |J|. Assume first J=∅. By Lemma 4.25, for every i∈ {1, . . . , t}there exist elements xi∈Giand subgroups U∅ i⊴oGisuch that φ∗(xiU∅ i) = {w∅}for a suitable w∅∈G. Moreover, by Remark 4.16, we may assume that Gj≤NG(U∅ i)for every j≤i. Hence, Lemma 4.29 produces a finite set σ(∅)of primes such that φ∗ I(Gi;U∅ i)⊆Nσ(∅)hw∅iG, so the claim follows for |J|= 0. Assume now that |J| ≥ 1and that for every J−(Jthere exist a finite set WJ−⊆φ∗(Gi), a finite set of primes σ(J−)and subgroups UJ− i⊴oGi,i /∈ I ∪ J−, such that φ∗ I∪J−(Gi;UJ− i)⊆Nσ(J−)hWJ−iG. For convenience, we also set UJ− i=Giif i∈J−, so that UJ− iis defined for all i /∈ I. Let WJ=SJ−WJ−, ρ=SJ−σ(J−)and Vi=TJ−UJ− ifor all i /∈ I, so that, by (4.4), we have 90 φ∗ I∪J−(Gi;Vi)⊆NρhWJiGfor every J−(J. Furthermore, by factoring out NρhWJiG, we may assume that φ∗ I∪J−(Gi;Vi) = 1 (4.8) for every J−(J. Moreover, taking into account Remark 4.16 we may further assume that Viis invariant under the conjugacy action of Gjfor every j≤i. Write J={j1, . . . , jn}with j1<· · · < jn, and for every i∈J, fix a set Siof coset representatives for Viin Gicontaining the identity. Write Sj1× · · · × Sjn={s1, . . . , sm} with sℓ= (sℓ,j1, . . . , sℓ,jn)for ℓ∈ {1, . . . , m}. Denote Vi=Gifor i∈ I. Since 1∈Sifor every i, we have φ∗ J(Si;Vi)6=∅, so applying Lemma 4.25 we obtain elements xi∈Viand subgroups Ui⊴oVisuch that φ∗ J(Si;xiUi)takes a single value. Actually, since 1 = φ∗ J(1; xiUi)⊆φ∗ J(Si;xiUi), we have φ∗ J(Si;xiUi)=1. Thus, for every ℓ∈ {1, . . . , m}, we either have φ∗ J(sℓ,i;xiUi) = ∅or φ∗ J(sℓ,i;xiUi) = 1.(4.9) We may assume xito be a good representative of the coset xiUiand therefore, if Jdoes not contain neither inor i+ 1, then (|xi|,|xi+1|)=1. Also, by Remark 4.16 we may further assume that Uiis invariant under the conjugacy action of Gj for every j≤i. Let J0=∅, and for r∈ {1, . . . , n}, let Jr={j1, . . . , jr}. We also write j0= 0 for convenience. We will show that for every r∈ {0, . . . , n}, there exists a finite set of primes τ(r)such that φ∗ Jr(sℓ,i;Y(r) i)⊆Nτ(r)for every ℓ∈ {1, . . . , m}, where Y(r) i=                Giif i≥jr, i ∈ I ∪ J, Uiif i > jr, i 6∈ I ∪ J, xiUiif i < jr. Notice that right now we are not using Y(r) jr, but it will be convenient to have it defined for later. We argue by reverse induction on r∈ {0, . . . , n}; assume first r=n. Since jr6∈ I, we deduce from (4.7) that (|sℓ,jr|,|Gjr−1|) = (|sℓ,jr|,|xjr+1|) = 1. Thus, for all ℓ∈ {1, . . . , m}, we obtain from (4.9) and Lemma 4.29 a finite set 91 of primes τ(r, ℓ)such that φ∗ Jr(sℓ,i;Y(r) i)⊆Nτ(r,ℓ). Defining τ(r) = Sm ℓ=1 τ(r, ℓ), we obtain φ∗ Jr(sℓ,i;Y(r) i)⊆Nτ(r)for every ℓ∈ {1, . . . , m}. Hence, we assume r≤n−1. By induction, we know that there exists a finite set of primes τ(r+ 1) such that φ∗ Jr+1 (sℓ,i;Y(r+1) i)⊆Nτ(r+1) (4.10) for every ℓ∈ {1, . . . , m}. The inductive step will be divided in two phases. We will first show that φ∗ Jr(sℓ,i;Y(r+1) i)⊆Nτ(r+1) (meaning that the only difference from (4.10) is position jr+1). In order to obtain this, we have to substitute in the jr+1-th position first sℓ,jr+1 , and then sℓ,jr+1 Ujr+1 for all ℓ∈ {1, . . . , m}. We will then conclude the inductive step by proving that there exists a finite set τ(r)of primes such that φ∗ Jr(sℓ,i;Y(r) i)⊆Nτ(r)for every ℓ∈ {1, . . . , m}. We begin by noting that Y(r+1) i≤Vifor every i6∈ I ∪ Jand that Ujr+1 ≤Vjr+1 , so (4.8) yields φ∗ Jr(sℓ,i;e Yi)⊆Nτ(r+1),(4.11) where e Yi=Y(r+1) iif i6=jr+1 and e Yjr+1 =Ujr+1 . As we chose the sets of representatives Sjin such a way that the identity is contained in them, for every ℓ∈ {1, . . . , m}, either sℓ,jr+1 is trivial or |π(sℓ,jr+1 )|= 1, so in particular sℓ,jr+1 is a good representative. Thus, by (4.10) and (4.11), we deduce from Lemma 4.22 that φ∗ Jr(sℓ,i;Yi)⊆Nτ(r+1), where Yi=Y(r+1) iif i6=jr+1 and Yjr+1 =sℓ,jr+1 Ujr+1 . Since this holds for every ℓ∈ {1, . . . , m}, and since Gjr+1 =Ss∈Sjr+1 sUjr+1 , we obtain φ∗ Jr(sℓ,i;Y(r+1) i)⊆Nτ(r+1), as we wanted. Now using (4.7) and Lemma 4.29, we conclude exactly as in the case r=nthat there exists a finite set τ(r)of primes such that φ∗ Jr(sℓ,i;Y(r) i)⊆Nτ(r)for every ℓ∈ {1, . . . , m}. This completes the reverse induction on r. In particular, for r= 0, it follows that φ∗ J(Gi;Ui)⊆Nτ(0), so this, in turn, concludes the inductive step on |J|, and the claim is proved. Finally, taking Jin such a way that I ∪ J={1, . . . , t}, we obtain a finite set of primes σ(J)and a finite set W⊆φ∗(Gi)such that φ∗(G1, . . . , Gt)⊆Nσ(J)hWiG, as desired. Recall that if Gis a prosolvable group of Fitting height k+ 1, there exist some pronilpotent subgroups U0, . . . , Uksatisfying Proposition 4.15. We remark that φand φ∗were defined with variables {xi|i= 1, . . . , t}for a generic positive integer t. Since we now want to apply the previous results to 92 the subgroups U0, . . . , Uk, we will set t=k+ 1 and we will write φ(Ui−1)for φ(U0, . . . , Uk)and φ∗(Ui−1)for φ((U0× · · · × Uk)∩ C), where Cis defined as in (4.3). Lemma 4.32. Let G=U0· · · Ukbe as in Proposition 4.15 with Uk=δ∗ k(G) abelian, and assume |δ∗ k{G}| <2ℵ0. Let g∈φ∗(Ui−1). Then, there exists a finite normal subgroup N⊴Gsuch that g∈N. Proof. Write g= [x0, . . . , xk], where xj∈Ujfor all jand (|xℓ|,|xℓ+1|)=1for all ℓ∈ {0, . . . , k−1}. By Lemma 4.9, [x0, . . . , xj]is a δ∗ j-value for every j∈ {0, . . . , k}. In particular x:= [x0, . . . , xk−1]is a δ∗ k−1-value. Let Hbe the minimal Hall subgroup of δ∗ k(G)containing xk, so that (|x|,|H|)=1, again by Lemma 4.9. Since, again, [x, h]is a δ∗ k-value for every h∈H, the set K:= {[x, h]|h∈H} has less than 2ℵ0values, and, since His abelian and normal in G, it follows that Kis actually a closed subgroup of G. In particular, Kis finite, so every element of Khas finite order. Thus, we deduce from Lemma 3.17 that the set S:= S{kG|k∈K}is finite, and therefore N=hSiis finite by Dietzmann’s Lemma (see Lemma 14.5.7 of [73]). We are now ready to prove the strong conciseness of δ∗ kin prosoluble groups of Fitting height k+ 1. Proposition 4.33. Let Gbe a prosoluble group of Fitting height k+ 1. Assume that |δ∗ k{G}| <2ℵ0. Then δ∗ k(G)is finite. Proof. In view of Lemma 4.12, we may assume that δ∗ k(G)is abelian. Thus, we can take U0, . . . , Uk≤Gas in Proposition 4.15, so that G=U0· · · Ukwith Uk=δ∗ k(G) abelian. We claim that for every family of subgroups Gi−1≤Ui−1with i∈ {1, . . . , k+1} such that Gj≤NG(Gi)for j≤i, we have |φ∗(Gi−1)|<∞. We argue by induction on |I|, where I={i∈ {1, . . . , k + 1} | |π(Gi−1)|=∞}. If |I| = 0, then Lemma 4.30 gives the result since for every finite set W⊆ φ∗(Gi−1), the normal subgroup hWiGis finite by Lemma 4.32, and since, by definition, Nσ= 1 for every finite set of primes σ. Suppose thus |I| ≥ 1. Then, Lemma 4.30 produces a finite set of primes σand a finite set W⊆φ∗(Gi−1)such that φ∗(Gi−1)⊆NσhWiG. Observe that by induction, for every j∈ I, we have |φ∗ {j}(Hi−1;Gi−1)|<∞, where Hi−1is the Hall σ-subgroup of Gi−1, and therefore Nσis finite by Lemma 4.32. Again by Lemma 4.32, hWiGis also finite, and the claim follows. 93 When studying actions of groups on trees, we often need to restrict to minimal invariant subtrees, whose existence is guaranteed by the following lemma, which is Proposition 2.4.12 of [70]. Lemma 5.4. If Gis a pro-Cgroup acting on a pro-Ctree Γ, then there exists a minimal G-invariant pro-Csubtree ∆of Γ. If ∆contains more than one vertex, then it is unique. The action of Gon a pro-Ctree Γis irreducible if Γhas no proper G-invariant subtrees. From now on, for every subset Sof a group Gacting on a pro-Ctree T, we denote by TSthe minimal pro-Csubtree on which hSi ≤ Gacts. Similarly to the abstract case, when elements commute we can obtain some additional information on their action. Lemma 5.5. Let Gbe a pro-Cgroup acting faithfully on a pro-Ctree T. 1. Let g, h ∈Gbe such that hnormalises hgi, then hleaves Tginvariant and in particular, if [g, h] = 1 then Tg=Th. 2. Let S={g1, . . . , gk}be a set of elements such that the action of each gi, i∈ {1, . . . , k}, is elliptic. If [gi, gj] = 1 for every i, j ∈ {1, . . . , k}, then there exists a vertex of Tfixed by the whole set S. Proof. Part (1) follows immediately by observing that h·(Tg) = Thgh−1⊆Tg. We first prove part (2) for two elements g1, g2∈G. If both g1and g2are elliptic, consider the subtrees Tg1and Tg2fixed by g1and g2respectively; by (1) we have that Tg1is a non-empty pro-Csubtree invariant under the action of g2. By Corollary 4.1.9 of [70], g1fixes a vertex of Tg2, hence Tg2∩Tg1is not trivial. Applying the case k= 2 to each pair, we have that Tgi∩Tgj6=∅and giand gj fix pointwise the intersection for every i, j ∈ {1, . . . , k}so we can apply Lemma 5.3 to the set {Tg1, . . . , Tgk}and conclude that Ti∈ITi6=∅and each gifixes this intersection, thus (2) follows. Let ∆ = (V(∆), E(∆)) be a graph. We set m∈∆if m∈V(∆) or m∈E(∆). A finite graph of pro-Cgroups (G,∆) over a finite abstract graph ∆is a collection of pro-Cgroups G(m)for each m∈∆, and continuous monomorphisms ∂i: G(e)−→ G(di(e)) for each edge e∈E(∆),i∈ {0,1}. We only work with finite graphs of pro-Cgroups, in the sense that the graph ∆is finite, but it is possible to define an analogous concept for graphs of pro-Cgroups over profinite graphs ∆ (see Chapter 6 of [70]). A graph of groups is reduced if edge groups corresponding to edges that are not loops are properly contained in adjacent vertex groups. 100 Definition 5.6 (Pro-Cfundamental group).Given a finite graph of pro-Cgroups (G,∆), we define its pro-Cfundamental group G= Π1(G,∆) as follows. Fix a maximal subtree Dof ∆; then Gis a pro-Cgroup, together with a collection of continuous homomorphisms νm:G(m)−→ G(m∈∆) and a continuous map E(∆) −→ G, denoted e7→ te(e∈E(∆)), such that te= 1 if e∈E(D), and such that (νd0(e)∂0)(x) = te(νd1(e)∂1)(x)t−1 e∀x∈ G(e), e ∈E(∆); that satisfies the following universal property: whenever we have • a pro-Cgroup H, • a collection of continuous homomorphisms βm:G(m)−→ H,(m∈∆), • a map e7→ se(e∈E(∆)) with se= 1 if e∈E(D), and •(βd0(e)∂0)(x) = se(βd1(e)∂1)(x)s−1 e∀x∈ G(e), e ∈E(∆), then there exists a unique continuous homomorphism δ:G−→ Hwith δ(te) = se (e∈E(∆)) such that for each m∈∆the diagram G δ  G(m) νm << y y y y y y y y βm"" E E E E E E E E H commutes. It was proven in [84] that this definition does not depend on the choice of the maximal subtree D, moreover the existence and uniqueness of this group is proven in Proposition 6.2.1 and Theorem 6.2.4 of [70]. One can construct the fundamental group of a graph of pro-Cgroups by iterating two operations, namely pro-Camalgamated products and pro-CHNN extensions, denoted by G1qHG2and HNN(G1, H, f)respectively, and where G1and G2are pro-Cgroups, H≤G1, and f:H→H′≤G1is an isomorphism. Both of these 101 constructions are defined by means of a universal property and can be obtained as a certain pro-Ccompletion of the abstract amalgamated product and HNN extension of the corresponding groups. We refer to Sections 9.2 and 9.4 of [72] for the precise definitions and basic properties. It is important to remark that, contrary to the abstract case, the factors G1and G2(resp. the base group G1) do not necessarily embed into G1qHG2(resp. HNN(G1, H, f)). Whenever they embed, the amalgamated product (resp. HNN extension) is said to be proper. Some necessary and sufficient conditions for proCamalgamated products and HNN extensions to be proper were described in Theorem 9.2.4 and Proposition 9.4.3 of [72]. We remark that properness is assured if if the amalgamated subgroup His a virtual retract of G1and G2(as G1and G2would induce the full pro-Ctopology on Hand the hypothesis of Thm 9.2.4 in [72] hold in this case). Abstract Bass-Serre theory relates fundamental groups of graphs of groups with groups acting on trees. Such a relation is true for the pro-Ccase assuming that the action on a pro-Ctree is cofinite and not true in general. Namely given a fundamental group of a graph of pro-Cgroups (G,∆), there is a natural pro-Ctree Ton which it acts. The construction of this tree, called the standard pro-Ctree, is described in Chapter 6 of [70]. The converse is true only for the cofinite action. If the fundamental group of the graph of pro-Cgroups is a pro-Camalgamated product G=G1qHG2or a pro-CHNN extension G=HNN(G1, H, f), then each vertex stabiliser Gvof a vertex vis a conjugate of G1or G2(or of G1if G=HNN(G1, H, f)) and each edge stabiliser Geis a conjugate of H. Abstract Bass-Serre theory is extremely useful for studying the structure of subgroups of fundamental groups of graphs of groups. The same is true for the pro-Cversion of Bass-Serre theory, and the main tool is Theorem 7.1.7 of [70]. We state the applications of these results to the case when the group acting on the pro-Ctree is a pro-Camalgamated product or HNN extension. As usual, we denote by b ZC=Qp∈π(C)Zpthe pro-Ccompletion of Zfor any set of primes π(C). Theorem 5.7. Let Kbe a subgroup of a proper free amalgamated pro-Cproduct G=G1qHG2of pro-Cgroups. Then one of the following holds: 1. K≤gGig−1for g∈Gand i∈ {1,2}; 2. Khas a non-abelian free pro-psubgroup Pfor a certain p∈π(C)such that P∩gGig−1= 1 for all g∈Gand i∈ {1,2}; 3. there exists a subgroup H0⊴K(which is the kernel of the action of Kon TK) that is contained in a conjugate of Hand such that K/H0is solvable 102 and isomorphic to a projective group Zσo Zρ(σ, ρ ⊆π(C)with σ∩ρ=∅) or ZσoCn(with σ⊆π(C)and Cna finite cyclic group). In the last case, it can be a profinite Frobenius group or, if Cn=C2and 2∈σ, an infinite dihedral pro-σgroup. Theorem 5.8. Let Kbe a subgroup of a proper pro-CHNN extension G= HNN(G1, H, f). Then one of the following holds: 1. K≤gG1g−1for g∈G; 2. Khas a non-abelian free pro-psubgroup Pfor p∈π(C)such that P∩ gG1g−1= 1 for all g∈G; 3. there exists a subgroup H0⊴K(which is the kernel of the action of Kon TK) that is contained in a conjugate of Hand such that K/H0is solvable and isomorphic to a projective group Zσo Zρ(σ, ρ ⊆π(C)with σ∩ρ=∅) or ZσoCn(with σ⊆π(C)and Cna finite cyclic group). In the last case, it can be a profinite Frobenius group or, if Cn=C2and 2∈σ, an infinite dihedral pro-σgroup. A useful remark is that, in the third case of the previous theorems, H/H0is torsionfree if and only if it is isomorphic to Zσo Zρ. In this case, as this is a projective group, we have that H∼ =H0o(Zσo Zρ). Finally, we record the following observation. Lemma 5.9. Let G=G1qHG2be a proper amalgamated pro-Cproduct of two pro-Cgroups G1and G2and let Tbe the standard pro-Ctree associated with this splitting. Let g1, . . . , gkbe a sequence of elliptic elements such that [gi, gi+1]=1 for all i∈ {1, . . . , k −1}. Then there are some vertices v1, . . . , vk∈V(T)(not necessarily distinct) such that g1∈Gv1and gi∈Gtifor each ti∈[vi−1, vi]. Proof. By Lemma 5.5 there exists a vertex vistabilized by every pair of commuting elements gi, gi+1 for every i∈ {1, . . . , k −1}. Define vkto be any vertex stabilized by gk. In this setting, gistabilizes both vi−1and vi, hence it stabilizes the whole subtree [vi−1, vi]by Corollary 4.1.6 of [70]. 5.2 Basics on pro-CRAAGs The aim of this section is to describe basic properties pro-CRAAGs. The abstract version of the definitions and results that we discuss can be found, for example, in [18]. 103 Let Γ=(V(Γ), E(Γ)) be an undirected finite graph without double edges or loops, where V(Γ) and E(Γ) are the set of vertices and edges respectively. A subgraph ∆<Γis called full if for all e∈Γwith d0(e), d1(e)∈∆we have that e∈∆. Notice that full subgraphs are uniquely determined by the subset of vertices V(∆) of V(Γ). Definition 5.10 (Right-angled Artin pro-Cgroups).The right-angled Artin pro-C grouppro-CRAAG (pro-CRAAG for short) GΓis the pro-Cgroup given by the pro-Cpresentation GΓ=hV(Γ)|[u, v] = 1 if and only if uand vare adjacent in Γi. We recall some standard terminology. Definition 5.11 (Canonical Generators).The generators associated with the vertices of Γare called canonical generators and, abusing the notation, we denote them with the same letter as the corresponding vertex. Definition 5.12 (Standard subgroups).A subgroup of GΓis called a standard subgroup if it is the subgroup generated by a subset V′⊆V(Γ). If Γ = ∅, by convention we set GΓto be the trivial subgroup. Abusing the notation, if S⊆V(Γ), we denote by GSthe standard subgroup generated by the full subgraph generated by S. We begin by stating some properties of standard subgroups. Lemma 5.13. Let GΓbe a pro-CRAAG. Then: 1. GΓis the pro-Ccompletion of the abstract RAAG G(Γ); 2. the standard subgroup generated by a subset of vertices V′⊆V(Γ) is the pro-CRAAG G∆generated by the full subgraph ∆⊆Γdetermined by V′; 3. the standard subgroups of GΓare retracts; 4. the intersection of standard subgroups is a (possibly trivial) standard subgroup. Proof. For every group G, we denote by b Gits pro-Ccompletion. 1. Follows from the pro-Cpresentation (see Definition 5.10). 2. In the abstract case, the subgroup of G(Γ) generated by V′is exactly G(∆), see for example Corollary 2.11 of [50]. As this subgroup is a retract of G(Γ), the pro-Ctopology of G(Γ) induces on it the full pro-Ctopology, so the pro-Csubgroup hV′i ≤ GΓis \ G(∆), that by (1) coincides with G∆. 104 3. The map pr∆:GΓ→G∆whose restriction to G∆is the identity and such that pr∆(v)=1for every v∈V(Γ) rV′is surjective. Since by (2) G∆is a subgroup of GΓ, we have that pr∆is a retraction onto G∆. 4. Consider two standard subgroups G∆, GΛof GΓ. By (3), a non-trivial element gof GΓis in G∆∩GΛif and only if pr∆(prΛ(g)) = g, but this composition of maps corresponds exactly to pr∆∩Λ(g), and therefore G∆∩GΛ= G∆∩Λ. It follows from the pro-Cversion of Theorem 9.2.4 of [72] that, if His a retract of two groups G1and G2, then a pro-CG1qHG2is a proper pro-Camalgamated product. Similarly, it follows from Theorem 9.4.3 that pro-CHNN-extension HNN(G1, H, f)is proper if His a retract of G1. As standard subgroups of a RAAG are retracts, we deduce the following. Corollary 5.14. Let GΓbe a pro-CRAAG. If GΓis a pro-Camalgamated product G1qHG2or a pro-CHNN extension HNN(G1, H, f)with G1, G2, H, f(H)standard subgroups of GΓ, then the free product with amalgamation or HNN extension is proper. We now want to define the notion of support of an element, but we first begin by proving that this concept is well-defined. Lemma 5.15. Let GΓbe a pro-CRAAG and let g∈GΓ. Then there exists a unique minimal standard subgroup containing g. Moreover there exists an element hin the conjugacy class of gwhose corresponding minimal standard subgroup is contained in each standard subgroup containing conjugates of g. Proof. The unique minimal standard subgroup containing gis the intersection of all the standard subgroups containing it, and this intersection is still a standard subgroup by Lemma 5.13. Suppose now that ∆1,∆2are full subgroups of Γsuch that g∈G∆1and gt∈G∆2for t∈GΓ. We claim that there exists s∈GΓsuch that gs∈G∆1∩∆2. Indeed let pr∆1be the retraction of GΓto G∆1and define s=pr∆1(t). Then gs=pr∆1(gt)∈G∆1∩∆2. In order to prove the lemma it suffices to apply this observation to the lattice of full subgraphs of Γcontaining a conjugate of g. Notice that if g= 1 we have that g∈G∅and by convention, the standard subgroup generated by the empty set is the trivial group. 105 Definition 5.16 (Support of an element).Let gbe an element of a (pro-C) RAAG GΓ. The support α(g)of gis the set of canonical generators of the unique minimal standard subgroup of GΓcontaining g. In view of Lemma 5.15, in any conjugacy class there exists an element gsuch that α(g)⊆α(gt)for every t∈G, in this case we say that gtis an element of minimal support among its conjugates. Definition 5.17 (Links and stars).Let gbe an element of a (pro-C) RAAG GΓ. The link Link(g)of gis the set of vertices of Γrα(g)that are adjacent to each of the vertices in α(g). If vis a canonical generator, we denote by Star(v)the full subgraph generated by Link(v)∪v. Remark 5.18. If v∈V(Γ), we can split GΓas a pro-CHNN extension as GΓ=HNN(GΓr{v}, GLink(v), id)(5.2) with stable letter v, and by Corollary 5.14 this is a proper pro-CHNN extension. It follows that if gis an element with minimal support among its conjugates and v∈α(g), then Theorem 5.8 guarantees that its action on the standard pro-Ctree Tassociated with this splitting is hyperbolic. Abstract right-angled Artin groups are torsion-free, but the pro-Ccompletion of torsion-free groups is not always torsion-free (even the profinite completion as shown in [54],[19]). However, in the case of pro-CRAAGs this is true. Theorem 5.19. Pro-CRAAGs are torsion-free profinite groups. Proof. A pro-CRAAG is the pro-Ccompletion of the corresponding (abstract) RAAG. In [30], the authors proved that abstract RAAGs are residually (finitely generated torsion-free nilpotent), and hence the pro-Ccompletion of a RAAG embeds in a direct product of the pro-Ccompletions of finitely generated torsionfree nilpotent groups. By Theorem 4.7.10 of [72] the profinite completion b Nof a finitely generated torsion-free nilpotent group Nis torsion-free. But b N=Qpb Np is the direct product of the pro-pcompletions and the pro-Ccompletion of Nis the direct product Qp∈π(C)Np. Hence the pro-Ccompletion of Nis torsion-free. 5.3 Direct product decomposition of pro-CRAAGs Our goal is to show that the direct product decomposition of a pro-CRAAG is determined by the defining graph. More precisely GΓ≃A1×A2, where A1and A2are non-trivial pro-Cgroups, if and only if Γis a join, see Theorem 5.22. 106 Lemma 5.20. Let GΓbe a pro-CRAAG and let g∈GΓbe an element with minimal support among its conjugates. Then, the centraliser of gis contained in the standard subgroup generated by Link(g)∪α(g). In particular, if g=vis a standard generator, then CG(v) = GStar(v)=hvi × GLink(v). Proof. Suppose towards contradiction that there is an element hcommuting with gwhose support is not contained in Link(g)∪α(g). Then there exists v∈α(h) such that v /∈Link(g)∪α(g). Denoting by G0=GΓr{v}and by A=GLink(v)and using Remark 5.18, we have that the group GΓsplits as a proper HNN extension of the form GΓ=HNN(G0, A, id) where the action by conjugation of von Ais trivial. Notice that from the assumption on h, we have that h /∈G0. We next study the action of gand hon the standard pro-Ctree Tassociated with this splitting. Notice that g∈G0and so gis elliptic. However, gcannot belong to any edge stabiliser. Indeed, otherwise, there would exist an element t∈GΓsuch that gt∈A and in this case, since ghas by assumption minimal support, it would follow from Lemma 5.15 that α(g)⊆α(gt)⊆Link(v)and so v∈Link(g)contradicting the choice of v. Since gcannot be in any edge stabiliser, we conclude that gonly fixes the vertex vstabilised by G0, i.e. Tg={v}. From Lemma 5.5 (1), hhas to leave Tg={v}invariant and, in particular, hfixes v. Then hbelongs to G0, a contradiction. Lemma 5.21. Suppose a pro-CRAAG G=GΓdecomposes as a direct product GΓ=A1×A2of non-trivial groups. Then for each canonical generator v∈Γ, at least one factor Aiis contained in GStar(v). Proof. Let vbe a canonical generator. Since α(v) = {v}, by Lemma 5.20 we have that CG(v) = GStar(v)=hvi × GLink(v). Suppose that v=a1·a2where ai∈Ai,i= 1,2. Since CG(v) = CA1(a1)×CA2(a2) and ai∈CAi(v), from the description of the centraliser CG(v), we deduce that ai=veia′ ifor ei∈Zπ(C)and a′ i∈GLink(v). Since v=a1·a2, we have that ei6= 0 for either i= 1 or i= 2; without loss of generality assume e16= 0. Let tbe an element such that t−1a1thas minimal support among its conjugates, we can assume t∈A1because A2⊆CG(a1). Applying Lemma 5.20 we have tA2t−1=A2⊆CG(a1) = tCG(t−1a1t)t−1⊆tGα(t−1a1t)×GLink(t−1a1t)t−1. Notice that by Lemma 5.15 α(t−1a1t)⊆α(a1)⊆Star(v)and, since v∈α(t−1a1t), the definition of link implies that Link(t−1a1t)⊆Star(v). Overall, we conclude that A2⊆GStar(v). 107 We are now ready to fully characterize when a pro-CRAAG splits as a direct product. We recall that a graph is a join if and only if there is a non-empty subgraph ∆Γsuch that for each v∈∆and each w∈Γr∆,v, w are adjacent. Theorem 5.22. Let GΓbe a pro-CRAAG. Then GΓhas a non-trivial direct product decomposition if and only if Γis a join. In particular, each factor in a direct product decomposition of GΓis a standard subgroup. Proof. The analogous result for abstract RAAGs is classical (see for example Corollary 2.15 in [50]). From the abstract result and Lemma 5.13, it is straightforward that whenever Γis a join, then GΓsplits as a direct product. We now want to prove the converse implication. By Lemma 5.21 for each canonical generator v, at least one among A1or A2is contained in GStar(v). Let Γ1⊆V(Γ) be the set of canonical generators vsuch that A1<Star(v)and Γ2= Γ rΓ1. Then, for each canonical generator v∈Γ2, since by definition of Γ2 we have that A16<Star(v), by Lemma 5.21 again we conclude that A2≤GStar(v). For i= 1,2define ∆i⊆Γsuch that G∆i=Tv∈ΓiGStar(v); by Lemma 5.13 G∆iis a standard subgroup and by definition it contains Aiand each v∈Γiis connected to each w∈∆i. In particular ∆iare non-empty graphs. Notice that if there is a canonical generator w∈∆1∩∆2, then wis by definition in the star of each vertex in Γiand so Γ1,Γ2<Star(w). Hence such a canonical generator w∈∆1∩∆2would be central and Γwould decompose as a join. For this reason we can assume that G∆1and G∆2are disjoint and since A1and A2generate G, so do G∆1and G∆2. Hence, we can decompose V(Γ) as the disjoint union of the (possibly empty) sets Γ2∩∆1,Γ1∩∆2and Λ = (Γ1∩∆1)∪(Γ2∩∆2). Since ∆iis non-empty for i= 1,2, then either Λ6=∅or Γ2∩∆1and Γ1∩∆2 are non-empty. If at least two of the sets are non-empty, then they define a join, because each vertex in a set is connected to each vertex in the other set, because each element in Γiis connected to each element in ∆ifor i= 1,2. We are left to consider the case when only Λis non-empty, so that Λ = V(Γ). In this case, each vertex in Γiis in ∆itoo and in particular they are connected to each other. It follows that Γi∩∆i= Γi= ∆iis a complete graph for i= 1,2. Since Ai≤G∆iand G∆iis abelian, so is Ai. Hence G=A1×A2is abelian and Γis a complete graph and a join. These results are in line with other properties of pro-CRAAGs that can be recognized from the abstract graph. For example, abstract RAAGs split as a 108 free product if and only if the underlying graph is disconnected, and Wilkes and Kropholler proved that the same is true for profinite RAAGs in [51]. Similarly, both abstract and pro-pRAAGs are coherent if and only if the underlying graph is chordal, see [76]. 5.4 Centralisers and normalisers of elements In this section, we describe explicitly the structure of centralisers of elements in pro-CRAAGs, obtaining a description similar to the one that Baudisch proved for abstract RAAGs in [12]. In a free pro-pgroup, centralisers of elements are cyclic. However, in the pro-Ccase, the situation is substantially different as the centraliser of an element does not need to be cyclic. Indeed, for example, the projective group Z3o Z2, with the generator of Z2, say a, acting on Z3by inversion, embeds in a free profinite group, so the centraliser of a2contains this solvable projective group. Theorem 5.23. Let G=GΓbe a pro-CRAAG and let g0∈G. Then there is an element gin the conjugacy class of g0such that its centraliser is of the form CG(g) = H1× · · · × Hs× hLink(g)i where: 1. α(Hi), α(Hj),Link(g)are all disjoint for i6=j; 2. Gα(g)=Gα(H1)× · · · × Gα(Hs); 3. Hiare projective pro-Cgroups; 4. if Gis pro-p,Hi=hhiiand g=hk1 1· · · hks s, for some ki∈Zp. Proof. We begin the proof with some reductions. If gis trivial, then V(Γ) = Link(g)and the result holds trivially, so we further assume g6= 1. Among the conjugates of g0, we choose an element gof minimal support among its conjugates, so that by Lemma 5.20 CG(g)is contained in the standard subgroup generated by Link(g)∪α(g). Hence, we can assume that V(Γ) = Link(g)∪α(g). In this case, we have from Theorem 5.22 that G=Gα(g)×GLink(g). Clearly GLink(g)≤CG(g), so it suffices studying the centraliser in the standard subgroup Gα(g)and then CG(g) = CGα(g)(g)×GLink(g). We further assume that α(g) = V(Γ). If Gis decomposable as a direct product GΓ=G1× · · · × Gs, then g=g1× · · · × gsfor gi∈Gi,i∈ {1, . . . , s}, and the 109 116 6 Abelian splittings of RAAGs In this section we will study how a pro-CRAAG can split as an amalgamated product or HNN extension over an abelian subgroup. In [35], Hull and Groves proved that an abstract RAAG splits over an abelian subgroup if and only if the underlying graph either has a separating complete graph or it is disconnected. This result extends a previous theorem of Clay [21], who proved it in the case of cyclic splittings. In the first section we prove that the same conditions are necessary and sufficient in order to have abelian splittings of pro-CRAAGs. We also point out that, if the underlying graph is connected, a conjugate of a standard subgroup is always contained in the abelian amalgamated subgroup. Describing all the abelian splittings of a group is in general difficult as some of them are not compatible with each other. In any case, there is a construction, called the JSJ decomposition, that encodes all the “universal” splittings of a group over a chosen class of subgroups. In the second section we give a description of JSJ decompositions in profinite groups, which is obtained following the approach of Guirardel and Levitt in [37]. In particular, we can define A-JSJ decompositions, meaning that we describe all splittings of a group when the amalgamated subgroups are in the class of groups A, and then relative (A,H)-JSJ decompositions, in the sense that every subgroup in the class His elliptic in the decomposition. In the third section we obtain a (A,H)-JSJ decomposition of pro-CRAAGs 117 in the case that Ais the class of abelian groups and His the class of procyclic subgroups generated by a canonical generator. The proof is constructive, in the sense that it inheritely provides an algorithm to obtain the aforementioned decomposition. In the last section we refine the relative decomposition in order to obtain a general A-JSJ decomposition. We conclude with an explicit example showing the algorithm beneath the construction of these decompositions. 6.1 Abelian splittings of profinite RAAGs The main goal of this section is to describe when and how a pro-CRAAGs splits over a pro-Cabelian group. We begin with two auxiliary lemmas. Lemma 6.1. Let G=GΓbe a pro-CRAAG associated with a connected graph Γ. Suppose that Gacts on a pro-Ctree Twithout a global fixed point, and that all canonical generators are elliptic. Then there exist two canonical generators v, w ∈V(Γ) such that (v, w)/∈E(Γ) and hv, widoes not stabilize any vertex of T. Proof. Let Tvbe the subtree of fixed points of a canonical generator v. If, by contradiction, Tv∩Tw6=∅for each couple of canonical generators v, w ∈V(Γ), by Lemma 5.3 there is a point contained in Tv∈V(Γ) Tvfixed by all the generators and so fixed by G, contradicting the hypothesis. This implies that there are at least two vertices v, w ∈V(Γ) such that hv, widoes not stabilize any vertex of T. Notice that such vertices cannot be adjacent by Lemma 5.5 (2). Lemma 6.2. Let GΓbe a pro-CRAAG over a connected graph Γacting on a pro-C tree Twith abelian edge stabilisers. Suppose that a canonical generator v∈GΓis hyperbolic, then: 1. Star(v)is a complete graph; 2. either V(Γ) = Star(v)or the set S:= {u∈Link(v)|Star(u)is not a complete graph} separates Star(v)rSand ΓrStar(v); 3. the standard subgroup generated by Sstabilizes an edge. Proof. 1. If there exists a single vertex adjacent to v, then the result holds. Suppose then that there exist two distinct vertices w1, w2∈Link(v). For each canonical generator wcommuting with vwe can restrict to the 118 minimal subtree T⟨v,w⟩on which the abelian subgroup hv, wiacts. By Theorems 5.7 and 5.8, the group hv, wiis a procyclic extension of the kernel of this action and since hv, wiis abelian of rank 2, there exists an element g=ab with a∈ hvi ≤ Gand b∈ hwi ≤ Gwith b6= 1 (as vis hyperbolic) in the kernel of the action, i.e. gfixes pointwise the minimal subtree T⟨v,w⟩. Pick now two elements gi=aibiwith i∈ {1,2}for a1, a2∈ hvi,b1∈ hw1i, b2∈ hw2isuch that b1, b2are not trivial and such that g1, g2stabilize pointwise Tv. By hypothesis g1, g2are contained in the abelian stabilisers of the edges of Tv. Let K=hg1, g2iand let fbe the retraction of Gonto the standard subgroup generated by w1, w2. The image f(K)≤G{w1,w2}is an abelian subgroup that contains b1and b2. The element b1is in the centraliser of b2and they are both with minimal support among their conjugates, so applying Lemma 5.20 this can happen only if w1∈Link(b2) = Link(w2), so w1, w2are adjacent and Star(v)is a complete graph. 2. Suppose V(Γ) 6= Star(v), as Γis connected we have that S={u∈Link(v)|Star(u)is not a complete graph} is non-empty. It is immediate to see that Sseparates the subgraphs generated by Star(v)rSand ΓrStar(v)because, as Link(v)is a complete graph by (1), each vertex in Star(v)rSis connected only to vertices in Star(v). 3. By Lemma 5.5(1), each vertex of Sfixes the subtree Tv, which contains at least an edge because vis hyperbolic. By (1), the action on Tof any element of Sis elliptic, and hence Tvis fixed pointwise by S. We are now ready to prove the main theorem of this section. Theorem 6.3. Let G=GΓbe a pro-CRAAG associated with a connected graph Γ. Then Gacts on a pro-Ctree with abelian edge stabilisers without a global fixed point if and only if either Γis a complete graph or Γhas a disconnecting complete graph. In the second case, there exists a disconnecting complete graph whose standard subgroup is contained in one edge stabiliser of T. Proof. The case when Γis a complete graph is clear: indeed, denoting by π= π(C), the pro-CRAAG GΓis isomorphic to Zn π, that splits as an HNN-extension 119 HNN(Zn−1 π,Zn−1 π, id). Similarly, if there is a complete graph Kthat disconnects Γ, i.e. ΓrK= Γ1∪Γ2with Γ1,Γ2disjoint subgraphs, then Gsplits as GΓ=GΓ1∪KqGKGΓ2∪K and so Gacts on the standard pro-Ctree associated with this splitting. Suppose now that Gacts on a pro-Ctree Twith abelian edge stabilisers. If there exists a hyperbolic canonical generator vof G, by Lemma 6.2 we know that either Γis complete or the set S:= {u∈Link(v)|Star(u)is not a complete graph}is a disconnecting complete graph contained in an edge stabiliser. We are left to the case when each canonical generator of Gis elliptic. By Lemma 6.1 there exist two vertices v, w ∈V(Γ) such that no vertex of Tis stabilized by both vand w. As each canonical generator acts elliptically, let tv, tw be two vertices of Tstabilized by vand wrespectively. Let S= [tv, tw]be the geodesic between these two vertices in T. By Lemma 6.1 Scontains at least one edge, and moreover there exists at least one edge of Sthat is not stabilised either by vor by w, as by collapsing the subtrees S∩Tvand S∩Twto a point (noticing that we chose v, w such that Tv∩Tw=∅), we would otherwise have Sto be disconnected.Define Kas a maximal (by the number of vertices contained) complete subgraph of Γcontained in an edge group of Sthat is not stabilized by either vor w, say e∈E(T). If Kis empty, let ebe any edge of S, not stabilized by vor w. It is important to notice that even if S might contain infinitely many edges satisfying the properties, Γis finite hence K is well defined. We claim that Kis a complete graph of Γthat disconnects the vertices vand w. Suppose by contradiction that it is not, then we could find a finite path p= (v, u1, . . . , uk, w)in Γ, such that no vertex of pis contained in K. By Lemma 5.9 there exist some vertices t1, . . . , tk+2 such that vstabilizes t1,uistabilizes the geodesic Si= [ti, ti+1]for i= 1, . . . , k, and wstabilizes [tk+1, tk+2]. Set t0=tvand Tk+3 =tw. In this setting, vstabilizes S0= [t0, t1]and wstabilizes Sk+1 = [tk+1, tw]. Furthermore, the union S′=Si∈{0,...,k+1}Siof the Siis a pro-p tree that contains tvand tw, hence it contains the whole [tv, tw]. In particular, S′contains e, so e∈[tj, tj+1]for some j∈ {0, . . . , k + 3}. By the choice of e, it cannot be stabilized by vor w, so there exists a vertex ujsuch that uj∈Ge. Now Geis an abelian pro-Csubgroup of Gthat contains ujand each vertex of K, but by Theorem 5.23 this is only possible if ujis adjacent to every vertex of K. By maximality of K,uj∈K, but this contradicts the fact that no element of the path pis contained in K. If we assumed Kto be empty, we have anyway proved that there is a vertex ujcontained in an edge stabiliser of Tcontradicting that 120 K=∅. This proves that the graph Kis a disconnecting complete graph contained in an edge stabiliser, as required. 6.2 JSJ decompositions Prerequisites In the previous section, we have characterised when a pro-CRAAGs admits a splitting over an abelian subgroup. Our next goal is to describe all the splittings of these groups over abelian subgroups. In the abstract case, the abelian splittings of a finitely generated group are encoded in a construction called the JSJ decomposition of a group. We develop this theory following the approach of Guirardel and Levitt in [37]. We show that it can be naturally extended to the pro-Cworld; for additional results and alternative definitions on the theory of JSJ decompositions see the references in [37]. Definition 6.4 (A-trees).For each class of pro-Cgroups Aclosed for subgroups and conjugation, we define an A-tree (T, G)as a pro-Ctree Twith an action of a pro-Cgroup Gsuch that each edge stabiliser is a group in the class A. We often denote the A-tree as Trather than (T, G)whenever the pro-Cgroup Gacting on it is clear by the context and we will say that an A-tree (T, G)is trivial if Tconsists of a single vertex stabilized by the whole G. We say that a subgroup Hof a pro-Cgroup Gis universally elliptic (for actions over A-trees) if the action of His elliptic over any A-tree (T, G)on which Gacts. Definition 6.5 (JSJ decompositions). • An A-tree (T, G)is universally elliptic if its edge stabilisers Ge≤Gare universally elliptic for actions on A-trees. • An A-tree (T, G)dominates another A-tree (T′, G)if the same group Gacts on both of them and the action of vertex stabilisers Gv,v∈Tis elliptic on T′too. • Two A-trees (T, G)and (T′, G)are equivalent if the same pro-Cgroup G acts on both of them and they dominate each other. An equivalence class of A-trees for this relation is said to be a deformation space. • The deformation space of the A-trees that are universally elliptic and that dominate any other universally elliptic A-tree on which Gacts is the JSJ deformation space and its elements are called the JSJ tree decompositions. 121 Notice that the deformation space is unique, but there might be many nonisomorphic tree decompositions of a pro-Cgroup G. Definition 6.6 (Rigid and flexible vertices).A vertex vof a JSJ-tree is said to be rigid if it is universally elliptic for the action on any A-tree (even if the tree is not universally elliptic) and flexible otherwise. Notice that if all vertex groups of an A-tree are rigid, then the A-tree is a JSJ tree, but the converse is not true, as the following example shows. Example 6.7. If G∼ =Zn ρfor n≥2,ρan arbitrary set of primes, the abelian JSJ decomposition is trivial. We claim that for each element g∈Gwe can produce an A-tree (T, G)such that the action of gon Tis hyperbolic. Consider a maximal procyclic subgroup Ccontaining g∈G. Any generator of a maximal procyclic group can be part of a basis of Zn ρ, so we can pick a complement B∼ =Zn−1 ρof Cin Gand write G=HNN(B, B, id)with a generator of Cas the stable letter. The standard pro-Ctree associated with this pro-CHNN extension is a vertex with a single loop and gis hyperbolic by construction. This proves that no edge group can be universally elliptic, hence there exists a single universally elliptic A-tree (T, G)on which Gacts, which is a tree Twith a single point. This is the JSJ decomposition of G, which has a single flexible vertex. Sometimes is convenient to study relative JSJ decompositions, which are defined as follows. Definition 6.8 (Relative JSJ Decompositions).Let Hbe an arbitrary family of subgroups of a pro-Cgroup G. An A-tree (T, G)is an (A,H)-tree if all the subgroups in the class Hare elliptic. An (A,H)-tree is an (A,H)-JSJ decomposition if it is universally elliptic for actions on (A,H)-trees and it dominates every other universally elliptic (A,H)-tree. We now turn our attention to the study of the JSJ-decomposition of a pro-C RAAG over abelian groups. Let G=GΓbe a pro-CRAAG over a finite connected graph Γ. From here on, we assume Ato be the class of abelian pro-Csubgroups of Gand Hto be the class of procyclic groups generated by canonical generators of G. We first construct by induction a decomposition of Gover abelian subgroups relative to H, and prove that it is actually an (A,H)-JSJ decomposition. We then refine this decomposition in order to obtain the A-JSJ decomposition of G. As we are interested in splittings over standard subgroups of disconnecting complete graphs, we first need some basic properties of splittings of this type. 122 Lemma 6.9. Let GΓbe a pro-CRAAG over a finite connected graph Γand K≤Γ be a complete subgraph of Γ. 1. If all cyclic subgroups generated by canonical generators in Kare universally elliptic for their action on A-trees, then the whole standard subgroup GKis universally elliptic for its action on A-trees. 2. If Kis a minimal disconnecting complete graph (in the sense that no proper subset of Kis a disconnecting complete graph), then the standard subgroup GKis universally elliptic for its action on A-trees. 3. If Star(v)is a complete graph for v∈V(Γ), then there exists an A-tree on which the action of vis hyperbolic. Proof. 1. Since by assumption Γis connected, this follows as a consequence of Lemma 5.5 (2). 2. Assume that Gacts on an A-tree (T, G)and suppose that there exists at least one hyperbolic canonical generator v∈V(K). By Lemma 6.2(1), we have that Star(v)is a complete graph. Since a complete graph does not have any disconnecting subgraphs, it follows that Γ6= Star(v). From the minimality of the disconnecting complete graph K, we have that the full subgraph Γ′generated by (V(Γ) rV(K)) ∪ {v}is connected and vis a disconnecting vertex of Γ′. In particular, there are two vertices w1, w2∈ V(Γ′)that are adjacent to vbut lie in different connected components of ΓrK. This contradicts the fact that Star(v)is a complete graph. Hence each canonical generator of Kmust be elliptic and by (1) the whole Kis elliptic. 3. It suffices to notice that the standard pro-Ctree associated with the splitting (5.2) has abelian edge stabilisers because Link(v)is a complete graph. We record the following graph theoretical observation. Lemma 6.10 (Disconnecting graphs of components).Let Γbe a finite connected simplicial graph. Let Kbe a disconnecting complete subgraph of Γand let {Γi| i∈ {1, . . . , m}} be the connected components of ΓrK. If K′is a disconnecting subgraph of Γj∪Kfor some j∈ {1, . . . , m}, then K′ is also a disconnecting subgraph of Γ. 123 Proof. Suppose on the contrary that ΓrK′is connected. Since Kis by assumption a disconnecting subgraph of Γ, it follows that Kis not contained in K′and so K\K′is nonempty. Since ΓrK′is connected and Kis disconnecting, for each vertex vin (Γj∪K)rK′there is a vertex w(v)in KrK′such that vand w(v)are connected by a path inside (Γj∪K)rK′. As Kis complete, there is an edge between any two vertices in K. It follows that any pair of vertices v, v′∈(Γj∪K)rK′are connected by the path which is the composition of the paths from vto w(v), the edge (w(v), w(v′)) and the path from w(v′)to v′. Since this path is in (Γj∪K)rK′, we have that (Γj∪K)rK′is connected, deriving a contradiction. 6.3 (A,H)-JSJ decomposition of pro-CRAAGs We first construct the (relative) abelian JSJ decomposition of pro-CRAAGs under the assumption that all the subgroups in the class H={hvi | v∈V(Γ)}of procyclic subgroups generated by canonical generators are elliptic. Theorem 6.11. Let G=GΓbe a pro-CRAAG associated with a connected abstract finite graph Γ. There is a (possibly trivial) decomposition of Gas a fundamental pro-Cgroup of a reduced finite tree of pro-Cgroups (G∆,∆) with the following properties: •vertex groups of (G∆,∆) are standard subgroups which are either abelian or their underlying graph does not contain any disconnecting complete subgraph; •each edge group of (G∆,∆) is a standard subgroup associated with a disconnecting complete subgraph Keof Γand, moreover, Keis a minimal (with respect to inclusion) disconnecting complete graph of a subgraph Γ′of Γ. Furthermore, the standard pro-Ctree associated with this decomposition is an (A,H)-JSJ tree decomposition (T∆, G)of G. Proof. We prove the statements by induction on the number of generators of the pro-CRAAG. Assume first that Γhas one vertex, i.e. G=Zπ(C). In this case, we consider the decomposition as a fundamental group of a graph of groups to be trivial, so ∆is a point and the associated group is Zπ(C). This decomposition satisfies the required conditions. Furthermore, since Gis a standard subgroup, by assumption it is elliptic and so the (A,H)-JSJ decomposition of Gis trivial and agrees with the decomposition as a fundamental group of a graph of groups. 124 Assume that we have already established the decomposition of every pro-C RAAG whose underlying graph has at most n−1vertices as a fundamental group of a graph of groups and that we have proved that the (A,H)-JSJ decomposition of Gis determined by the group decomposition as a fundamental group of a graph of pro-Cgroups satisfying the properties of the theorem. Let now Γbe a connected graph with nvertices, n≥2. Suppose first that Γ does not have any disconnecting complete subgraph. In this case, we consider the decomposition as a fundamental group of a graph of groups to be trivial and so ∆ has one vertex with corresponding group G. This decomposition satisfies the requirements. If Γis a complete graph, then G≃Zn π(C). Since by assumption, each canonical generator is elliptic, then by Lemma 6.9, the group Gstabilizes a point, and hence the (A,H)-JSJ decomposition is trivial and coincides with the decomposition of Gas the fundamental group of a graph of groups. If Γis not complete and does not have any disconnecting complete subgraph, then by Theorem 6.3 G cannot act non-trivially on an A-tree, so the (A,H)-JSJ decomposition is again trivial. Suppose now that Γhas a disconnecting complete graph. Let Kbe a disconnecting complete graph such that |V(K)|is minimal among disconnecting complete graphs. We first construct a splitting of Gas an amalgamated free product over the standard subgroup GK. Assume that ΓrKhas m≥2nontrivial connected components Γi, for i∈ {1, . . . , m}. In this case, we consider the splitting of Gas a pro-Camalgamated product of the form G= m a i=1 GKGK∪Γi. 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Algebra, 624:1–16, 2023. 139 140 Index canonical generators, 104 cell, 52 decomposition, 52 concise boundedly, 20 on normal subgroups, 30 strongly, 22 uniformly boundedly, 23 word, 19 condition C′(λ), 55 coprime commutator γ∗ kand δ∗ k, 72 simple, 72 deformation space, 121 diagram, 53 circular, 53 over a group, 53 reduced, 54 domination of A-trees, 121 elliptic element, 99 extension of w(N), 44 flexible vertex, 122 full subgraph, 104 geodesic of a pro-Ctree, 99 good representative, 84 graph of pro-Cgroups, 100 fundamental group of, 101 reduced, 100 hanging vertex, 127 hyperbolic element, 99 independent set of relations, 56 irreducible action, 100 isolated subgroup, 111 join graph, 108 JSJ decomposition, 121 (A,H)-, 122 relative, 122 law, 15 link of an element, 106 marginal subgroup, 15 minimal simple group, 94 O-pairs, 63 Outer commutator extension, 43 outer commutator height, 44 words, 15 period of rank i, 56 piece, 54 pro-CRAAG, 104 141