Fixed subgroups in free groups: a survey

Con empo a y Ma hema ics Fixed Subg oups In F ee G oups: A Su ey E. Ven u a Abs ac . This no e is a su ey o he main esul s known abou ixed sub- g oups o endomo phisms o ini ely gene a ed ee g oups. A his o ic poin o iew is aken, emphasizing he e olu ion o his line o esea ch, om i s beginning o he p esen ime. The a icle concludes wi h a sec ion con aining he main open p oblems and conjec u es, wi h some commen s and discussions on hem. 1. In oduc ion and no a ion The pu pose o his no e is o su ey wha is known abou ixed subg oups o ini ely gene a ed ee g oups. Desc ip ions, commen s and ema ks on he p oo s o known esul s will be p esen ed ins ead o he p oo s hemsel es, o which we e e he eade o he co esponding o iginal pape s. We will also ake he oppo uni y o e iew he his o y o his pa icula opic in G oup Theo y, which goes back o he 1970’s. Howe e , i is no ou in en ion o be exhaus i e in his su ey. We apologize o hose au ho s who con ibu ed o his line o esea ch and a e no ci ed he e. In sec ions 2-7 we will adop a his o ic poin o iew, desc ibing he esul s ch onologically as hey appea ed in he li e a u e, e en when some o hem imp o e olde ones. Finally, in sec ion 8, a lis o some conjec u es and open p oblems in his a ea o esea ch will be p o ided and discussed. The whole pape is abou ini ely gene a ed ee g oups. F om now on, Fnwill deno e a ee g oup o ank n≥0, while Fwill be used o deno e an a bi a y ee g oup (possibly wi h in ini e ank). I is well known ha e e y subg oup o a ee g oup is also ee (see, o example, Theo em I.8.4 in [11]). Bu , in gene al, i s ank can be la ge han he ank o he ambien g oup. In ac , i is easy o see ha he subg oup H=hb− ab , ∈Zio F2=ha, biis ee o coun ably in ini e ank. So, Fℵ0is a subg oup o F2. The e o e, e e y ee g oup o ini e o coun ably in ini e ank can be iewed as a subg oup o e e y o he , wi h he ob ious excep ion o he i ial g oup, and o he ee g oup wi h ank 1, which is he g oup o in ege s. Also, i is well known ha he in e sec ion o ini ely gene a ed subg oups o a ee g oup is again ini ely gene a ed (see [30] o Theo em I.8.8 in [11]), while i s ank can be o he o de o he p oduc o he wo anks. This beha io , which is comple ely di e en om wha happens in o he mo e classical algeb aic con ex s, mo i a ed a lo o esea ch dedica ed o compu e o bound he ank o se e al subg oups c 0000 (copy igh holde ) 1 2 E. VENTURA in di e en si ua ions. We a e in e es ed in he case in ol ing ixed subg oups o endomo phisms o Fn. The educed ank o a ee g oup F, deno ed ˜ (F), is max{0, (F)−1}, ha is, one less han he ank, excep o he i ial g oup whe e he educed ank coincides wi h he ank, which is ze o. So, a ee g oup has educed ank ze o i and only i i is cyclic. Le End(Fn) deno e he monoid o endomo phisms o Fn, and Au (Fn) he g oup o au omo phisms o Fn(so, Au (Fn) is he g oup o uni s o End(Fn)). Le Inj(Fn) deno e he se o injec i e endomo phisms o Fn, a submonoid o End(Fn) con aining Au (Fn). Finally, we deno e by Ou (Fn) he g oup o ou e au omo - phisms o Fn, ha is, Au (Fn) modulo he (no mal) subg oup o conjuga ions, also called inne au omo phisms. Fo each u∈Fn, we le uagain deno e he co e- sponding igh conjuga ion, u:Fn→Fn,x7→ xu=u−1xu. We le elemen s o End(Fn) ac on he igh o Fnand, i he e is no isk o con usion, we will omi he pa en hesis o he a gumen . Thus, xφ deno es he image o xunde φ, and xφ1φ2deno es (xφ1)φ2. Fo any S⊆End(Fn), le Fix Sdeno e he se consis ing o he elemen s o Fnwhich a e ixed by e e y elemen o S(wi h he con en ion ha Fix S=Fn when Sis emp y). Then, Fix Sis a subg oup o Fn, called he ixed subg oup o So he subg oup ixed by S. When Sis a single on, S={φ}, we simply w i e Fix φins ead o Fix {φ} o he ixed subg oup o φ. So, Fix S=∩φ∈SFix φ. Clea ly, i S⊆End(Fn) and Mis he submonoid o End(Fn) gene a ed by S hen Fix S= Fix M. Following he no a ion in oduced in [38], a subg oup H≤Fnis called endo- ixed i H= Fix S o some S⊆End(Fn). I Scan be chosen o lie in Inj(Fn) ( esp. Au (Fn)) we u he say ha His a mono- ixed ( esp. au o- ixed) subg oup o Fn. And i Scan be chosen o be a single on, S={φ} o some φ∈End(Fn) ( esp. φ∈Inj(Fn), φ∈Au (Fn)) we say mo e explici ly ha His a 1-endo- ixed ( esp. 1-mono- ixed,1-au o- ixed) subg oup o Fn. Le φ∈End(Fn). We will concen a e on hose esul s conce ning Fix φas a subg oup o Fnand, o ins ance, only a ew commen s will be made abou he nice ecen esul s on in ini e wo ds ixed by he ex ension o φ o he bounda y o Fn. Also, we will no be conce ned in gene al abou isola ed ixed wo ds, o abou simila esul s in non- ee g oups. The s udy o ixed subg oups in ee g oups began in 1975 wi h he pape [15] by J. Dye and P. Sco . Since hen, new esul s ha e been appea ing cons an ly. I is ag eed ha he main and deepes esul in his line o esea ch is he Bes ina- Handel Theo em, published in [2]. Fo his eason, he p esen pape is o ganized as ollows. In sec ion 2 we e iew he o iginal mo i a ions, as well as he i s pa ial esul s. In sec ion 3 we e iew he main esul s abou ixed subg oups ob ained in he pe iod 1982-1992, be o e he Bes ina-Handel Theo em. Sec ion 4 is en i ely dedica ed o he Bes ina-Handel heo y. Sec ion 5 is abou he esul s ob ained by using he Bes ina-Handel Theo em and by gene alizing i . Sec ion 6 conside s he maximal ank case, while sec ion 7 is dedica ed o he concep o ine ia, p obably he s onges p ope y cu en ly known o ixed subg oups o ini ely gene a ed ee g oups. Finally, sec ion 8 con ains a lis o conjec u es and open p oblems in his line o esea ch, as well as se e al commen s and discussions. FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 3 2. The Sco conjec u e In 1975, J. Dye and P. Sco published he pape [15]. Using a heo em o Ka ass–Pie owski–Soli a abou he s uc u e o ee-by- ini e g oups, he au ho s ob ained he ollowing esul : Theo em 2.1 (Dye -Sco , [15]).I Gis a ini e g oup o au omo phisms o a ee g oup F, hen Fix Gis a ee ac o o F. In pa icula , o e e y ini e o de au omo phism φ∈Au (Fn), we ha e (Fix φ)≤n. This pape can be conside ed as he s a ing poin o he line o esea ch abou ixed subg oups in ini ely gene a ed ee g oups. In [15], he au ho s men ioned ha i was no known i Fix Gis ini ely gene a ed o a bi a y subg oups G≤ Au (Fn). In iew o his esul , P. Sco conjec u ed ha his is he case a leas o single au omo phisms (i.e. when Gis cyclic). His main mo i a ion was he ollowing classical esul due o J. Nielsen: Theo em 2.2 (Nielsen, [41]).Le Gpbe he undamen al g oup o a closed o ien able su ace wi h genus p≥1, and le Hbe he subg oup ixed by some au omo phism o Gp. Then, ei he H=Gpo His ee wi h ank a mos 2p−1. Fu he mo e, i His no cyclic, hen Gphas a se o 2pgene a o s such ha some subse o i gene a es H. In 1975, no example was known o a 1-au o- ixed subg oup o Fnwi h ank bigge han n. So, one could also conjec u e ha (Fix φ)≤n o e e y φ∈ Au (Fn). The ini eness o his ank o he p e ious inequali y can also be s udied o endomo phisms ins ead o au omo phisms, and mo e gene ally, o a bi a y se s o endomo phisms. Soon, all hese s a emen s became gene ically e e ed o as he Sco conjec u e. And ime jus i ied i , since i is now known ha (Fix S)≤n o e e y subse S⊆End(Fn). In 1977, W. Jaco and P. B. Shalen published [34], whe e hey s udied he ixed and pe iodic subg oups o hose au omo phisms o he undamen al g oup o a compac connec ed su ace Tinduced by homeomo phisms o he su ace T. Such an au omo phism is called geome ic o T. In he case whe e Tis closed and o ien able, his p o ides an al e na i e and en i ely di e en p oo o Nielsen’s Theo em (using he ac ha , o hese pa icula su aces, e e y au omo phism o π1(T) is geome ic o T). Fo he cases whe e Thas bounda y, he Jaco-Shalen Theo em gi es he ollowing co olla y: Theo em 2.3 (Jaco-Shalen, [34]).Le φbe an au omo phism o Fn. I φis geome ic ( o some compac connec ed su ace), hen Fix φis ei he cyclic o a ee ac o o Fn. In pa icula , (Fix φ)≤n. Howe e , his esul did no comple ely sol e he Sco conjec u e because, as J. S allings showed in [44], he e exis au omo phisms o ini ely gene a ed ee g oups which a e no geome ic ( o any compac connec ed su ace). Conc e ely, J. S allings in oduced he concep o PV-au omo phism o Fn:φ∈Au (Fn) is a PV- au omo phism when he absolu e alue o all he eigen alues o i s abelianiza ion a e less han 1, excep exac ly one which is la ge han 1. Wi h a simple homological a gumen , he showed ha no powe o any PV-au omo phism is geome ic. The simples such au omo phism is he one in F3=ha, b, cigi en by a7→ b,b7→ c, c7→ ab. Fu he mo e, J. S allings also conjec u ed ha any PV-au omo phism o 4 E. VENTURA Fnwi h n≥3 has i ial ixed subg oup. This was known as he S allings PV- conjec u e, and i p o ided new e idence in suppo o he Sco conjec u e. As a mo i a ion o his conjec u e, J. S allings p o ed (bu did no w i e) ha he ixed subg oup o a PV-au omo phism o Fnis con ained in he hi d e m in he lowe cen al se ies o Fn(see §1 in [46]). The ollowing s ep, in 1982, was he wo k [20] due o S. Ge s en (and published in 1984). He e, he au ho conside ed ano he amily o au omo phisms o Fn, wi h mo i a ions om one-dimensional geome y, i.e. om g aphs. Le Fnbe iewed as he undamen al g oup o a bouque Ywi h npe als. Fo e e y g aph Xand e e y maximal sub ee Tin X, one can collapse T o a poin and ob ain X/T, which is a bouque wi h as many pe als as he ank o ( he undamen al g oup o ) X. The e is also he na u al p ojec ion map X→X/T inducing an isomo phism a he undamen al g oup le el. An isomo phism φo Fn=π(Y, ∗) is called a change o maximal ee au omo phism (CMT-au omo phism o sho ) when he e exis s a ( ank n) g aph X, wo maximal sub ees T, T 0in X, iden i ica ions o X/T and X/T0wi h Y, and a e ex in Xsuch ha he co esponding isomo phisms ηT:π(X, )→π(X/T, ∗)≃Fand ηT0:π(X, )→π(X/T 0,∗)≃Fsa is y η−1 TηT0= φ. In [20], S. Ge s en sol ed he Sco conjec u e o CMT-au omo phisms p o ing he ollowing esul , using only combina o ial a gumen s: Theo em 2.4 (Ge s en, [20]).Le φbe an au omo phism o Fn. I φis a CMT-au omo phism, hen (Fix φ)≤n. Despi e being ini e o e e y n≥1, he amily o CMT-au omo phisms o Fn con ains in e es ing elemen s. Fo example, all Whi ehead au omo phisms and all Squie ’s skew-Nielsen au omo phisms o Fna e CMT-au omo phisms. In addi ion, S. Ge s en ga e a me hod such ha , when i e mina es, i com- pu es he ixed subg oup o a CMT-au omo phism. Howe e , his me hod is no eally an algo i hm because i can go in o an in ini e loop. The au ho analyzed some pa icula cases and p o ided he example gi en by he au omo phism o Fn=hx1, . . . , xni,n≥2, gi en by xi7→ xi+1x1 o i= 1, . . . , n −1 and xn7→ x1, which is simul aneously a CMT-au omo phism and a PV-au omo phism, and has i ial ixed subg oup. This was he i s in ini e amily o examples o which he PV-conjec u e was known o be ue. 3. Fixed subg oups be o e Bes ina-Handel In he same yea o i s publica ion, an addendum o [20] appea ed wi h mo e in o ma ion. Fi s , he au ho ga e a simple e ised e sion o his me hod o compu e ixed subg oups o CMT-au omo phisms. Then, he announced a p oo o he Sco conjec u e in gene al. This p oo was ou lined in [21] bu he de ails did no appea un il 1987 in he pape [22]. The idea in ol es s udying composi ions o Whi ehead au omo phisms. These au omo phisms gene a e Au (Fn), and hey a e CMT-au omo phisms so Ge s en’s me hod applies o hem. A e a ca e ul combina o ial analysis, he au ho deduces ha Fix φis ini ely gene a ed o e e y φ∈Au (Fn). Then, using Howson’s Theo em, he same can be said o ini ely gene a ed g oups o au omo phisms o Fn: Theo em 3.1 (Ge s en, [22]).Le Gbe a ini ely gene a ed g oup o au omo - phism o Fn. Then, Fix Gis ini ely gene a ed. FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 5 This sol ed he ques ion o iginally asked by P. Sco o he ini ely gene a ed case. Howe e , when one s udies he composi ion o CMT-au omo phisms, one loses con ol on he uppe bound o he ank o he ixed subg oup, and he a gumen gi en by S. Ge s en says only ha his subg oup is ini ely gene a ed. In [22], an uppe bound o (Fix φ) is gi en, bu i depends on he au omo phism φi sel and can be a bi a ily la ge, in con as wi h he uni o m bound (Fix φ)≤nknown a he ime o pe iodic, geome ic and CMT-au omo phisms o Fn. Du ing he pe iod 1982-1987, using comple ely di e en echniques, o he au- ho s ob ained independen p oo s o Ge s en’s Theo em, as well as se e al ex en- sions o i . And due o he nuances o ma hema ical publica ion, many o Ge - s en’s successo s saw hei wo k published be o e his pape appea ed. Fo example, see [24], [25], [26], [10] and [46], e iewed below. In 1983, J. S allings published [45], whe e some powe ul g aph- heo e ical echniques, e.g. he pull-back o g aphs, we e i s in oduced and used o gi e al e na i e p oo s o classical esul s on ee g oups. A he end o [45], he au ho en u ed a gene aliza ion o he Sco conjec u e, which has become known as he S allings conjec u e o he equalize conjec u e. Gi en wo g oups G, G0, and wo homomo phisms ϕ, φ:G→G0, he equalize o ϕand φ, deno ed Eq(ϕ, φ), is he maximal subg oup o Gwhe e bo h homomo phisms coincide, Eq(ϕ, φ) = {x∈G|xϕ =xφ}. The S allings conjec u e says ha i ϕ, φ:F→F0a e wo homomo phisms o ee g oups, Fis ini ely gene a ed, and ϕis injec i e, hen Eq(ϕ, φ) is also ini ely gene a ed. No e ha his conjec u e is s onge han he Sco conjec u e since, i ϕis an isomo phism, hen Eq(ϕ, φ) = Fix φϕ−1. No e also ha his s a emen is easily seen o be alse when bo h mo phisms ha e non- i ial ke nel. Conside he ee g oup F2=ha, biand he wo endomo phisms ϕ, φ gi en by aϕ = 1, bϕ =b and aφ = 1, bφ =b−1; clea ly, he equalize Eq(ϕ, φ) is he no mal closu e o ain F2, which has in ini e ank. In [22], ano he example o his ac is p oduced bu wi h he common image o he equalize being no ini ely gene a ed. Assuming ha ϕmus be injec i e, one can es ic he a en ion o he case whe e Fis a subg oup o F0and ϕis he inclusion. Then, Eq(ϕ, φ) = Fix φ, and φ:F→F0is an a bi a y homomo phism. This se ing is mo e gene al han wo king wi h endomo phisms o F0, because he exis ence o an ex ension o φ o an endomo phism o he whole o F0is no equi ed. Simila ly o he case o ixed subg oups, one can also ask (o conjec u e) whe he he ank o he equalize o ϕ, φ:F→F0is bounded abo e by ha o F, when ϕis injec i e. O e en mo e, one can ask i (Eq(S)) ≤ (F) o e e y se So mo phisms om F o F0con aining a leas one injec i e mo phism. As we will see below, he S allings conjec u e was p o ed some yea s la e , while hese s onge ques ions a e s ill open oday. We will gene ically e e o all o hem as he equalize conjec u e. A he 1983 AMS Summe mee ing, J. S allings poin ed ou ha he g aph cons uc ed and used by S. Ge s en o p o e he ini eness o he ank o 1-au o- ixed subg oups o Fn, could also be cons uc ed using he amilies o su aces desc ibed by he 3-dimensional model cons uc ed by Whi ehead in he 1930’s (see [54] and [55]). R. Golds ein and E. Tu ne de eloped his idea in a se ies o wo pape s, [24] and [25]. As is said in he in oduc ion o he i s pape , i so happens 6 E. VENTURA ha he use o h ee dimensions p o ides a eedom ha makes i easie o desc ibe and cons uc Ge s en’s g aph, and o p o e i s main p ope ies. In he i s o hese wo pape s, published in 1984, hey ga e an al e na i e p oo o Ge s en’s Theo em and hey imp o ed Ge s en’s uppe bound o he ank o he ixed subg oup o a gi en au omo phism o Fn. In he second one, which appea ed in 1985, hey wen u he using he same echniques and p o ed he S allings conjec u e o he case o wo injec i e homomo phisms. Fu he mo e, in 1986, ano he pape by R. Golds ein and E. Tu ne appea ed, [26]. He e, he au ho s imp o ed hei p e ious esul , sol ing comple ely he S allings conjec u e: Theo em 3.2 (Golds ein-Tu ne , [26]).Le Fand F0be wo ee g oups wi h F ini ely gene a ed. I ϕ, φ:F→F0a e wo mo phisms and ϕis injec i e hen Eq(ϕ, φ)is ini ely gene a ed. In his case, he p oo is qui e simple, sho and independen o he p e ious ones, and i p o es a s onge esul . In he si ua ion whe e Fis a subg oup o F0 and ϕis he inclusion, he au ho s p o ide a pa icula desc ip ion o he co e ing space Xo a bouque co esponding o he inclusion Fix φ≤F. Then, hey choose an app op ia e o ien a ion on he edges o Xand use i o show ha his g aph has ini e ank, and hen so does Fix φ. In 1987, D. Coope published [10] wi h ano he p oo o Ge s en’s Theo em. This ime, he me hods used a e o a opological and dynamical na u e. They a e inspi ed by wo k o W. Thu s on abou su ace g oups. The idea was o ex end a gi en au omo phism o Fn o a homeomo phism o i s end comple ion ˆ Fn, a compac me ic space whe e Fnis dense. The au ho hen showed ha he ixed poin se o his ex ension is ini ely gene a ed in a ce ain opological sense, which implies he ini e gene a ion o he subg oup ixed by he o iginal au omo phism. In pa inspi ed by D. Coope ’s wo k, h ee mo e pape s appea ed, de eloping new concep s and esul s in ol ing in ini e wo ds. These wo ks made i clea ha a close connec ion be ween he appa en ly di e en p oo s o [10] and [26] exis s. Two o hem a e [31] and [32], bo h published in 1990. In [32], W. Im ich and E. Tu ne conside ed an a bi a y homomo phism φ:F→F0 om a subg oup Fo an a bi a y ee g oup F0, o F0. By ex ending Golds ein-Tu ne echniques, hey ob ained new uppe bounds o he ank o Fix φ, and o he ank o he ixed poin se (`a la Coope ) o an injec i e φ. These bounds a e exp essed in e ms o a Nielsen educed basis o F. The hi d pape men ioned abo e is [6], by M.M. Cohen and M. Lus ig, and published in 1989. In his in e es ing wo k, he au ho s ex ended he Golds ein- Tu ne me hod by imposing a ec o ield (i.e. a p e e ed o ien a ion o he edges o he i s ba ycen ic subdi ision) on he g aph cons uc ed in [26]. They in oduced he concep s o a ac i e edges, epulsi e edges, a ac ing ixed in ini e wo ds and a ac ing ixed poin s a in ini y, and in es iga ed he esul ing dynamics. In [6], he au ho s ga e wo explici o mulas o he exac alue o he ank o he ixed subg oup o a gi en au omo phism o Fn, one in geome ic e ms and he o he in a comple ely algeb aic con ex . In gene al, i is no easy o make use o hese o mulas o conc e e examples and, also, hey do no seem o gi e enough in o ma ion o de i e a uni o m uppe bound o he ank o he 1-au o- ixed subg oups o Fndepending only on n. Howe e , hey applied hei me hod FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 7 o ob ain an explici algo i hm o compu ing he ixed subg oup o a posi i e au omo phism o Fn. Recall ha a posi i e au omo phism is one o which he e exis s a basis o Fnwhose images a e posi i e wo ds, see sec ion 6 in [6]. They also ob ained some use ul esul s o which he ollowing a e a sample: Theo em 3.3 (Cohen-Lus ig, [6]).Le {x1, . . . , xn}be a basis o he ee g oup Fn, and le φ∈Au (Fn). I ideno es he numbe o occu ences o xiin ( he no mal o m o ) xiφo he o m xiφ= ·xi·w, wi h , w ∈Fnand he abelianiza ion o wbelonging o he image o Id −φab, hen (Fix φ)≤Pn i=1 i. In pa icula , i o e e y i= 1, . . . , n,xidoes no occu in xiφ, hen Fix φ= 1. Theo em 3.4 (Cohen-Lus ig, [6]).Le {x1, . . . , xn}be a basis o he ee g oup Fn, and le φ∈Au (Fn)be an au omo phism such ha x1φ, . . . , xnφa e posi i e wo ds. I no wo d xiφbegins o ends wi h xi, hen Fix φ= 1. Theo em 3.5 (Cohen-Lus ig, [6]).Le Φ∈Ou (Fn)be an ou e au omo phism o Fn. Then, (Fix φ)≤1 o e e y φ∈Φexcep o hose belonging o a ini e numbe o conjugacy classes o elemen s in Φ. The e is also ano he ex ension o Ge s en’s Theo em in a di e en di ec ion, due o J. S allings in 1987. The pape [46], de eloped simul aneously o [25] o Golds ein-Tu ne , p o ided a p oo o he ollowing a ia ion o he S allings conjec u e: Theo em 3.6 (S allings, [46]).Le Fand F0be wo ee g oups wi h F ini ely gene a ed. Le ϕ, φ:F→F0be wo homomo phisms wi h ke ϕ= ke φ. Then he (common) image o Eq(ϕ, φ)unde ϕo φis ini ely gene a ed. In pa icula , i ϕ and φa e injec i e hen Eq(ϕ, φ)is ini ely gene a ed. He e, J. S allings de eloped his dyads and he g aphic echniques o olds and ladde a achmen s. He used hese echniques o build an ex ension o Ge s en’s p oo sui able o s udy he mo e gene al si ua ion ea ed in his pape . So, [46] ocuses on he g aphic and combina o ial poin o iew, wi hou in ol ing ei he 3- dimensional a gumen s o in ini e wo ds (al hough i con ains an appendix ela ing he dyads o Heegaa d diag ams in 3-mani olds). Addi ionally, [46] con ains a lis o he main ques ions and conjec u es abou his opic a ha ime. This lis con ains he abo e men ioned ques ions abou n being a uni o m uppe bound o he ank o 1-au o- ixed, 1-endo- ixed, au o- ixed o endo- ixed subg oups o Fn, o he co esponding ones o he equalize o a amily o wo o mo e homomo phisms (one o hem being injec i e). The e is also ano he in e es ing ques ion ela ing hese p oblems wi h subg oups o Fn×Fn. Fu he mo e, one inds he pape s [27] and [28], published by A.H.M. Hoa e in 1988 and 1990, espec i ely. He e, he Whi ehead 3-dimensional model ([54] and [55]), Ge s en’s g aph ([22]), he Golds ein-Tu ne cons uc ion, ([24], [25]) and he dyads o S allings ([46]) a e conside ed and ela ed o each o he . A he same ime, he au ho o e ed some a ia ions and ex ensions o all hese cons uc- ions, as well as a combina o ial p oo o he classical Whi ehead cu e ex lemma using hem. Howe e , no new esul s abou ixed subg oups we e gi en. I is also wo h men ioning he ex emely sho pape [48] published by S. Thomas in 1988. Using p e iously known esul s om A.G. Howson, J. S allings 8 E. VENTURA and J. McCool, as well as Ge s en’s Theo em, he au ho de ini ely closed he o igi- nal ques ion o Dye -Sco in [15] by d opping he hypo hesis on Gin Theo em 3.1: Theo em 3.7 (Thomas, [48]).Le Gbe an a bi a y g oup o au omo phisms o Fn. Then, Fix Gis ini ely gene a ed. 4. The wo k o Bes ina-Handel The nex s ep in his his o y is he deepes and mos ele an one in he whole line o esea ch abou ixed subg oups in ee g oups. In 1988 M. Bes ina and M. Handel announced a p oo o he s onge e sion o he Sco conjec u e, namely he uppe bound n o he ank o 1-au o- ixed subg oups o Fn: Theo em 4.1 (Bes ina-Handel, [2]).Le φbe an au omo phism o Fn. Then, (Fix φ)≤n. This esul p o ides he i s (and he bes possible) uni o m uppe bound o (Fix φ), in con as wi h he p e iously known ones, all o hem being ei he no uni o m o alid only o special amilies o au omo phisms. This esul appea ed in published o m, wi h a delay o almos ou yea s, in he i y-page pape [2] in he Annals o Ma hema ics in 1992. Again, some applica ions and imp o emen s o he Bes ina-Handel Theo em appea ed in p in be o e [2] was published. This excellen and dense wo k de eloped a new and powe ul g aphic heo y o ep esen a bi a y au omo phisms o Fn, and ended up wi h a p oo o Sco con- jec u e as a consequence. I has a highly opological la o and i is no o iously ha d o ead. I was inspi ed in he classical wo k o W. Thu s on on homeomo phism o su aces. W. Thu s on, ollowing a p og am s a ed many yea s be o e by J. Nielsen, ga e a classi ica ion o homeomo phisms o a compac su ace S, up o iso opy. Fu he mo e, he cons uc ed a pa icula ly e icien ep esen a i e o e e y iso opy class, ha is, o e e y elemen o he mapping class g oup o he su ace, MCG(S). This ep esen a i e can always be chosen o be ei he o ini e o de , o pseudo- Anoso , o educible (i.e. cons uc ible in a simple way, om homeomo phisms o simple su aces, which a e ei he o ini e o de o pseudo-Anoso ). Thu s on’s wo k was i s desc ibed in he p ep in [49] in 1976, h ee yea s la e in [16] ( he p oceedings o a semina held a O say in 1978), and inally published wel e yea s la e in [50]. See also [5] o an in oduc ion, and [3] o an algo i hmic p oo o Thu s on’s Theo em, based on [2]. Mapping class g oups o compac su aces, MCG(S), and ou e au omo phism g oups o ee g oups, Ou (Fn), a e simila in many ways. In [2], M. Bes ina and M. Handel de eloped a p ojec analogous o Thu s on’s, eplacing su aces by g aphs, and homeomo phisms by homo opy equi alences, a so o one-dimensional e sion o Thu s on’s wo k. Thei main esul was he cons uc ion o a g aph Zand an e icien homo opy equi alence β:Z→Z, inducing a p e iously gi en ou e au omo phism Φ ∈Ou (Fn) a he undamen al g oup le el. He e, e iciency consis s o some good p ope ies con olling he amoun o cancella ion appea ing in he images o single edges o Zunde i e a es o β. One has o hink o Fnas he undamen al g oup o he bouque Rn, a g aph wi h a single e ex and nedges. A ma ked g aph is a connec ed g aph Z oge he wi h a homo opy equi alence ρ:Rn→Z. Then, e e y homo opy equi alence FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 9 β:Z→Zo a ma ked g aph Zinduces an au omo phism o Fnup o conjuga ion, ha is, an ou e au omo phism Φ ∈Ou (Fn). I is hen said ha βis a opological ep esen a i e o Φ, o ha βinduces Φ. In he i s pa o [2], he au ho s analyzed he case o i educible ou e au o- mo phisms. An ou e au omo phism Φ ∈Ou (Fn) is said o be educible i he e exis ee ac o s G1, . . . , Gko Fnwhose conjugacy classes a e pe mu ed by Φ, and such ha G1∗ · · · ∗ Gkis s ill a ee ac o o Fn; o he wise, i is called i educible. See [13] o a classi ica ion o he simples i educible au omo phisms. M. Bes ina and M. Handel p o ed ha e e y i educible au omo phism can be opologically ep esen ed by a ain ack ha is, a homo opy equi alence β:Z→Zsuch ha βkis locally injec i e in he in e io o e e y edge (i.e., no cancella ion appea s in he i e a e images o e e y edge eβk, o e∈EZ, and k≥1). Using hese ain acks, hey p o ed he ollowing wo esul s: Theo em 4.2 (Bes ina-Handel, [2]).Le Φ∈Ou (Fn)be an i educible ou e au omo phism o Fn. Then, (Fix φ)≤1 o e e y φ∈Φ. Theo em 4.3 (Bes ina-Handel, [2]).Le Φ∈Ou (Fn)be such ha Φkis i educible o e e y k≥1. I he e exis a cyclic wo d ssuch ha sΦ = so sΦ = s−1, hen Φis geome ically ealized by a pseudo-Anoso homeomo phism o a compac su ace wi h one bounda y componen . The o me is a s onge e sion o he Sco conjec u e o i educible au o- mo phisms. The la e was quickly used by S. Ge s en and J. S allings o sol e he PV-conjec u e (see [23] and he nex sec ion). In sec ion 5 o [2], M. Bes ina and M. Handel ex ended hei heo y om i - educible o a bi a y au omo phisms o Fn. Any homo opy equi alence β:Z→Z de ines a na u al il a ion o Zby (no necessa ily connec ed) β-in a ian sub- g aphs, ∅=Z0< Z1<· · · < Z =Z, such ha Zi−1is a maximal p ope β-in a ian subg aph o Zi,i= 1, . . . , . The subg aph cl(Zi Zi−1) is called he i- h s a um. I is p o en ha an ou e au omo phism is i educible p ecisely when all i s opological ep esen a i es o e co e g aphs (i.e. ini e g aphs wi h- ou e ices o alence one) con aining no non- i ial β-in a ian o es s, con ain no non- i ial β-in a ian subg aphs. In his case, he co esponding il a ion has only one s a um. Now, hey in oduced he no ion o ela i e ain ack which, up o echnical de ails, is a na u al ela i iza ion o he no ion o ain ack, wi h espec o he p e ious il a ion. Wi h a conside able amoun o echnical wo k, he au ho s ended up wi h he esul ha e e y au omo phism o Fnadmi s a opological ep esen a i e which is a ela i e ain ack. Then, analyzing Nielsen pa hs, i.e. pa hs in he g aph ixed up o homo opy, hey inally ob ained he desi ed p oo o he Sco conjec u e. The p ocess o building such a good opological ep esen a i e is con olled by a ce ain ma ix. He e, he Pe on-F obenius Theo em (see Theo em 8.4.4 in [29] o II.1 in [14]) plays a cen al ole. A squa e non-nega i e in ege ma ix Mis called i educible when o e e y en y (i, j) he e exis s k≥0 such ha he (i, j)- h en y o Mkis posi i e; o he wise, Mis educible. Now, he Pe on-F obenius Theo em s a es ha e e y i educible ma ix Mhas he ollowing p ope ies: (i) i has a unique eigen alue wi h maximum modulus, (ii) his eigen alue is eal, i.e. he 16 E. VENTURA maximum- ank 1-au o- ixed subg oups o Fn. He e, Fab ndeno es he abelianiza ion o Fn, and o a, b ∈Fnwe w i e [a, b] = a−1b−1ab. Theo em 6.2 (Collins-Tu ne , [9]).Le Hbe a subg oup o Fn, and le m deno e he ank o he ( ee abelian) image o Hin Fab n(called he abelian ank o H). The ollowing a e equi alen : a) His a maximum- ank 1-au o- ixed subg oup o Fn, b) he e exis s a basis {x1, . . . , xn}o Fnsuch ha , se ing Fl=hx1, . . . , xli o 0≤l≤n, he e exis s a basis {y1, . . . , yn}o Hsuch ha o 1≤j≤ m,yj=xj, and o m+1 ≤k≤n,yk= [wk, xk] o some wk∈H∩Fk−1 no being a p ope powe o any elemen o Fn(so, in pa icula , wk6= 1). In his e en , {y1, . . . , yl}is a basis o H∩Fl, o 0≤l≤n. Fu he mo e, e e y au omo phism φ∈Au (Fn)wi h H≤Fix φis o he o m xj7→ xjand xk7→ w k kxk o some non-ze o in ege s k,1≤j≤m,m+ 1 ≤k≤n. The ypical example o a maximum- ank 1-au o- ixed subg oup is gi en by he ollowing au omo phism o F2=ha, bi:a7→ a,b7→ a b, whe e is an in ege . I s ixed subg oup is H=ha, [a, b]i=ha, b−1abi, and m= 1, excep when = 0 (in which case i is he whole F2). Howe e , o n≥3, he ixed subg oup o an au omo phism o he ype desc ibed in he las pa ag aph o he p e ious heo em can be bigge han H. Fo example, le be a non-ze o in ege and conside he au omo phism φ o F3=ha, b, cigi en by a7→ a,b7→ ab,c7→ a c. Fo 6= 1 we ha e Fix φ =ha, b−1ab, c−1aci, which has abelian ank m= 1. Bu Fix φ1is bigge since i also con ains b−1c, and has abelian ank m= 2 (in ac , Fix φ1=ha, b−1c, b−1abi=ha, b−1c, c−1aci). Addi ionally, in [9] he e is also an analogous esul desc ibing he ixed sub- g oups o au omo phisms o ee p oduc g oups, wi h maximal Ku oˇs ank. Finally, he pape [53], published in 1997 by E. Ven u a, also conside ed he case o maximal ank. The main esul in [53] was he ollowing: Theo em 6.3 (Ven u a, [53]).Among he s ic ly ascending chains o maxi- mum- ank 1-au o- ixed subg oups o Fn, he maximum leng h is exac ly n. The p oo uses he Collins-Tu ne desc ip ion o maximum- ank 1-au o- ixed subg oups o Fn. Essen ially, i is a g aphic p oo and i in ol es imme sions and co e ings o g aphs, as well as S allings oldings. Also, a simple g aphic p oo o an old esul om M. Takahasi was p o ided. Fo he pa icula case n= 2, and being Ha maximum- ank 1-au o- ixed subg oup o F2, a desc ip ion was gi en o all hose subg oups K≤F2sa is ying (K) = (H∩K) = 2. The main consequence de i ed om his esul was he ollowing heo em, which is clea ly only alid in he ank 2 case: Theo em 6.4 (Ven u a, [53]).Le Sbe a non-emp y se o non-iden i y en- domo phisms o F2such ha (Fix S) = 2. Then, S⊆Au (F2), and Fix S= ha, b−1abi o some basis {a, b}o F2, and Fix S= Fix φ o each φ∈S. In pa icula , his implies ha e e y g oup o au omo phisms G≤Au (F2) wi h (Fix G) = 2 is ei he i ial o in ini e cyclic. The pape [53] ends by showing ha , in he ank 2 case, he amilies o 1-au o- ixed, 1-endo- ixed, au o- ixed and endo- ixed subg oups do coincide. As men ioned abo e, in [40] an example is gi en o a 1-endo- ixed no 1-au o- ixed subg oup o Fn, o n≥3. FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 17 I is no known in gene al whe he he amilies o 1-au o- ixed and au o- ixed (o 1-endo- ixed and endo- ixed) subg oups o Fncoincide (see 8.9 and 8.10 below). Howe e , he pape [38] ( e iewed la e ) ga e a posi i e solu ion o his p oblem in he maximal ank case, as a consequence o he main esul he e. The esul is he ollowing: Theo em 6.5 (Ma ino-Ven u a, [38]).Le H≤Fnbe a subg oup o Fnwi h (H) = n. The ollowing a e equi alen : a) His a 1-au o- ixed subg oup o Fn, b) His a 1-mono- ixed subg oup o Fn, c) His a 1-endo- ixed subg oup o Fn, d) His an au o- ixed subg oup o Fn, e) His a mono- ixed subg oup o Fn, ) His an endo- ixed subg oup o Fn. 7. Ine ia and ixed subg oups In he monog aph [14], published in 1996 by W. Dicks and E. Ven u a, a new concep was in oduced: ine ia o subg oups o a ee g oup. The main esul in his wo k was he ine ia p ope y o mono- ixed subg oups o Fn, a s onge esul han he Sco conjec u e o hose subg oups. This esul is a pa ial solu ion o he unsol ed equalize conjec u e. A na u al pa icula case o ha conjec u e was he mo i a ion o look a he no ion o ine ia. The gene al equalize conjec u e says ha he equalize o wo homomo phisms ϕ, φ:Fm→Fn, one o hem being injec i e, has ank bounded by m. I , o ex- ample, ϕis injec i e hen he e is no lose o gene ali y in assuming ha Fmis a subg oup o Fnand ϕis he inclusion. In his case, we ha e only one homo- mo phism o conside , φ:Fm→Fn,Fm≤Fn, and Eq(ϕ, φ) = Fix φ. So, he equalize conjec u e is he Sco conjec u e gene alized o pa ial endomo phisms, i.e. homomo phisms om a ce ain subg oup o Fn, o Fn. The case Fm=Fnis he Sco conjec u e al eady sol ed by Bes ina-Handel. The simples s ep o conside nex is he case when φ:Fm→Fnex ends o an endomo phism ˜ φo Fn; in his case, Fix φ=Fm∩Fix ˜ φ. We can ese his si ua ion by aking φ∈End(Fn), and an a bi a y subg oup K≤Fn, and asking whe he (K∩Fix φ)≤ (K). This was he o iginal mo i a ion o he ollowing de ini ion. A subg oup H≤Fnis called ine when (K∩H)≤ (K) o e e y subg oup K≤Fn. O cou se, aking K=Fn, i H≤Fnis ine hen (H)≤n. Easy examples o ine subg oups o Fna e he cyclic ones, and he ee ac o s o Fn. No so ob ious is he ac ha e e y ank 2 subg oup o Fnis ine . This is because o wo k by G. Ta dos on he H. Neumann conjec u e. In [47] his conjec u e, i.e. ˜ (H∩K)≤˜ (H)˜ (K) o e e y H, K ≤Fn, was p o ed when one o he wo subg oups in ol ed has ank 2. This is p ecisely he same as saying ha ank wo subg oups o Fna e all ine . The main esul in [14] p o ided new examples o ine subg oups o Fn: Theo em 7.1 (Dicks-Ven u a, [14]).E e y mono- ixed subg oup o Fnis ine . In pa icula , (Fix S)≤n o e e y S⊆Inj(Fn). This esul can be hough o bo h as he nex s ep owa ds he solu ion o he gene al Sco conjec u e (be o e, i was only known o single endomo phisms, 18 E. VENTURA and his was he i s esul o amilies), o as a pa ial solu ion o he gene al equalize conjec u e. Wi h a simple obse a ion one can see ha he amily o ine subg oups o Fnis closed unde a bi a y in e sec ions ( his is clea o ini e in e sec ions, and a s anda d a gumen on a descending chain o subg oups p o es i in he in ini e case, see Co olla y I.4.13 in [14]). Then, i is enough o p o e Theo em 7.1 o 1-mono- ixed subg oups. This is wha i is done in [14]. The wo k [14] con ains a comple e e o mula ion and ex ension o he Bes i- na-Handel heo y, imp o ing i in he ollowing ou di ec ions. Though he pape [2], he e is a opological la o in many a gumen s (con- inui y, limi s, densi y, Can o se s, e c). Howe e , one obse es ha his is no essen ial because he inal esul , as well as all he in e media e ones, a e o a pu ely combina o ial and algeb aic na u e. One o he goals o [14] was o e o mula e he en i e heo y in [2] in a comple e algeb aic se ing, making anspa en i s al- geb aic na u e. The only essen ial in o ma ion ha is ca ied by a pa h in a g aph, is he educed sequence o edges c ossed by he pa h. So, i u ns ou o be mo e con enien o hink o a g aph jus as a combina o ial objec , and con inuous maps be ween g aphs as o mal maps sending edges (and educed pa hs) o educed pa hs. A good way o modelling his is he ca ego ical language o g oupoids. Associa ed o any g aph Z, one has he undamen al g oupoid πZ, which mus be hough o as he se o pa hs in Zmodulo educ ion, and wi h conca ena ion, which is a pa ially de ined ope a ion. Then, con inuous maps simply ansla e in o g oupoid mo phisms. One o he ad an ages o his se ing is ha he no ion o igh ening in [2] disappea s comple ely in he new language. This is because, when igh ening a gi en pa h pin a g aph, one ob ains ano he pa h p0, bu hey bo h gi e exac ly he same elemen o πZ. Ano he ad an age is ha he undamen al g oup o Z a a e ex is now iewed as a subg oup o he g oupoid πZ. So, passing o he undamen al g oup does no in ol e a mo e o a di e en con ex , i is jus he es ic ion o a ce ain subg oup. This way, con inuous maps and wha hey induce a he undamen al g oup le el a e uni ied in a common con ex . Also, he no ion o opological ep esen abili y can jus be exp essed as a simple equi alence ela ion in he con ex o g oupoid mo phisms. Ano he impo an poin o his e o mula ion o [2] is ha i a oids alking abou he in a ian il a ion ∅=Z0< Z1<· · · < Z =Z. Following an idea o Gabo iau-Le i -Lus ig, he en i e a gumen o [2] is ew i en jus looking a he op s a um, and hen a guing by induc ion on he educed ank o he unde lying (no necessa ily connec ed) g aph. This simpli ies conside ably he echnical de ails, since one only has o wo k wi h a single in a ian subg aph (ins ead o he whole il a ion) and a single ma ix and Pe on-F obenius eigen alue and eigen ec o (ins ead o one o each s a um in he il a ion). While ansla ing in o he algeb aic se ing, he echnical de ails we e a anged in such a way ha he su jec i i y p ope y o he au omo phism is inally used nowhe e. Hence, he en i e a gumen , and so he inal conclusion, wo ks o in- jec i e endomo phisms and no jus o au omo phisms o Fn. This is he i s gene aliza ion p o ided. Howe e , he cons uc ion is a om being alid in he p esence o a non- i ial ke nel. FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 19 The ou h and main imp o emen is he gene aliza ion o he heo y o p o ide a p oo o he ine ia o ixed subg oups. This was achie ed by mixing he Bes ina- Handel heo y wi h he pull-back echnique de eloped by J. S allings in [45]. Gi en an injec i e endomo phism φ∈Inj(Fn), he au ho s o [14] conside ed he g aph Xand he map X→Zcons uc ed in sec ion 6 o [2], and inducing he inclusion Fix φ≤Fn(see sec ion 4). This is whe e M. Bes ina and M. Handel used he good g aphical p ope ies o his g aph and g aph map o show ha ˜ (X)≤˜ (Z) (and consequen ly, (Fix φ)≤n). Ins ead, in [14] he au ho s conside ano he a bi a y ini e connec ed co e-g aph Yand map Y→Z( his ep esen s an a bi- a y di e en subg oup K≤Fn). Then, hey look a he pull-back o hese wo g aphical maps (p e iously a anged o be imme sions), say W→Z. By [45], he la e imme sion ep esen s he in e sec ion K∩Fix φ. And he good echnical p ope ies we e enough no only o show ha ˜ (X)≤˜ (Z), bu also o p o e ha he inequali y passes h ough he pull-back diag am. In his way, i was p o ed ha ˜ (W)≤˜ (Y), which means (K∩Fix φ)≤ (K) o an a bi a y K≤Fn. This is p ecisely he ine ia o he ixed subg oup. In ac , in [14] he non-connec ed e sion o he p e ious esul was p o ed. The analysis o he algeb aic meaning o he di e en componen s o Wyields a inal inequali y in ol ing he sum o educed anks o subg oups o he o m Ky∩Fix(φx−1), whe e x anges o e a se o ep esen a i es o Reidemeis e classes o φ, and, o e e y x,y anges o e a se o double cose ep esen a i es o K Fn/Fix(φx−1) (see Theo em IV.5.5 in [14] o he de ails). This non-connec ed e sion o ine ia was codi ied using se s ac ed on by Fn. Ano he in e es ing ac abou [14] is ha all he de ails le o he eade in [2] we e me iculously e i ied in he new con ex . Finally, ano he concep is in oduced in [14] by elaxing he ine ia condi ion. A subg oup H≤Fnis called comp essed when (H)≤ (K) o e e y o he subg oup H≤K≤Fn. Clea ly, ine subg oups a e comp essed, and i is easy o see ha e ac s a e also comp essed. I is no known whe he comp essed subg oups a e ine . I is e en unknown whe he e ac s a e ine . These p oblems a e ela ed wi h some conjec u es and open ques ions abou ixed subg oups (see sec ion 8). A e [14], wo mo e pape s appea ed in he li e a u e using, o building upon, he concep o ine ia. The i s is [1], due o G. Be gman and published in 1999. Among o he esul s, [1] con ains he ollowing heo em, which de ini ely sol ed he emaining piece o he gene al Sco conjec u e: Theo em 7.2 (Be gman, [1]).Le S⊆End(Fn)be an a bi a y amily o endomo phism o Fn. Then, (Fix S)≤n. So, he Sco conjec u e was comple ely sol ed by 1999. Howe e , a new p ob- lem came up na u ally om [14]. In despi e o Be gman’s esul , one can ask i Fix φis ine when φhas non- i ial ke nel. As a as we know, his is s ill an open p oblem oday (see sec ion 8 o addi ional commen s). Be gman’s p oo o Theo em 7.2 is based on a educ ion o he injec i e case by using his main esul : Theo em 7.3 (Be gman, [1]).Le Gbe a ini ely gene a ed o sion- ee g oup, Lan a bi a y g oup, and conside he ee p oduc G∗Land he p ojec ion π:G∗ 20 E. VENTURA L→G. I σ1, σ2:G→G∗La e wo g oup- heo e ic sec ions o π, hen Eq(σ1, σ2) is a ee ac o o G. The p oo o his esul used esul s o S allings-Swan and a de ailed analysis o suppo s o de i a ions. Also, a he end o [1], he e is an in e es ing lis o ques ions. Some o hem will be discussed in he nex sec ion. I is also in e es ing o ema k ha W. Dicks and M. Dunwoody ga e a di e en and mo e gene al p oo o Theo em 7.3. In [12], hey p o ed exac ly he same esul , bu wi hou using he o sion- ee hypo hesis. This new p oo makes use o he Almos S abili y Theo em, o p o ees and also o some new olding sequence echniques. Ano he wo k elying on he ine ia p ope y is [38], published in 2000 by A. Ma ino and E. Ven u a. He e i was conjec u ed ha e e y au o- ixed subg oup is 1-au o- ixed. The au ho s we e no able o p o e his ac (which is s ill open oday), bu hei main esul p o ides suppo o his conjec u e: Theo em 7.4 (Ma ino-Ven u a, [38]).Le S⊆End(Fn)and le Mbe he submonoid o End(Fn)gene a ed by S. Then, he e exis s φ∈Msuch ha Fix S is a ee ac o o Fix φ. The p oo was comple ely algeb aic, and conside ed i s Scon aining wo ele- men s, one o hem injec i e, hen S⊆Inj(Fn) and, inally, he gene al case using Be gman’s esul . Theo em 7.4 na u ally aises he ques ion o whe he a ee ac- o o a 1-endo- ixed subg oup is again 1-endo- ixed. This is ob iously ue when n= 2 bu i is no ue o n≥3, and he ollowing coun e example is p o ided in [38]. By Theo em 6.2, he subg oup ha, [a, b],[a, c]i=hai∗h[a, b],[a, c]i o F3=ha, b, ciis 1-au o- ixed. Bu he au ho s p o ed ha any endomo phism o F3 ixing [a, b] and [a, c] is o ced o ix a. So, h[a, b],[a, c]iis a ee ac o o a 1-au o- ixed subg oup o F3, which is no e en endo- ixed. The example can be easily gene alized o an a bi a y n≥3. As men ioned in sec ion 6, Theo em 7.4 also has consequences o he maximal ank case (see Theo em 6.5). The pape [38] also in oduced he concep o au o- ixed closu e. The au o- ixed closu e o a subg oup H≤Fnis Hc= Fix(Au H(Fn)), whe e Au H(Fn) is he subg oup o Au (Fn) o hose au omo phisms φsuch ha H≤Fix φ. As in classical Galois heo y, H≤Hcbu he inequali y can be s ic . Fo example, he p e ious example can be used o say ha he au o- ixed closu e o H=h[a, b],[a, c]iin F3 is Hc=ha, [a, b],[a, c]i. In gene al, he ela ionship be ween any gi en subg oup H o Fn, and i s au o- ixed closu e is qui e obscu e. 8. Conjec u es and open p oblems A lis o in e es ing p oblems and conjec u es ha s ill emain open is gi en in his inal sec ion. We u he add some commen s and discussions on hem. Some o he ques ions men ioned he e a e also asked and discussed in he las sec ion o [14] and [1]. Fi s o all, i is comple ely sa is ac o y ha he gene al Sco conjec u e, he o iginal mo i a ion o his line o esea ch, is al eady comple ely sol ed. The wo k o Bes ina-Handel, Im ich-Tu ne , Dicks-Ven u a and Be gman, shows ha FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 21 (Fix S)≤n o any a bi a y se So endomo phisms o Fn. Howe e , as usually happens in ma hema ics, e en when he o iginal p oblem is comple ely sol ed, he line o esea ch is no exhaus ed because o he in e es ing ques ions came up na u ally om he de eloped wo k. Fo example, i is no known i ixed subg oups o endomo phisms o Fna e necessa ily ine . Conjec u e 8.1 (Ine ia conjec u e).E e y endo- ixed subg oup o Fnis ine . Equi alen ly, e ac s o Fna e ine . Discussion. Le S⊆End(Fn). In [14] i is p o ed ha Fix Sis ine when S⊆Inj(Fn). So, he open p oblem is abou endomo phisms wi h non- i ial ke nel. In his case, Be gman showed in [1] ha (Fix S)≤n o a bi a y S, gi ing suppo ing e idence o he conjec u e. The se o ine subg oups o Fnis closed unde a bi a y in e sec ions. So, i is enough o conside 1-endo- ixed subg oups. Le φ∈End(Fn). By a esul o E. Tu ne [51], φ es ic s o an isomo phism o he s able image Fnφ∞, which is a e ac o Fncon aining Fix φ. So, Fix φis ine as a subg oup o a e ac o Fn. Hence, he conjec u e is equi alen o saying ha e ac s a e ine . The e is a ai ly explici desc ip ion o e ac s o Fn(see exe cise 15 in sec ion 3.2 o [36]), bu his does no seem o help in sol ing his p oblem. 2 Ques ion 8.2 (Be gman, [1]).Le R, H ≤Fnbe wo ini ely gene a ed sub- g oups o Fn. Is i always ue ha i Ris a e ac o Fn hen R∩His a e ac o H? Discussion. The mo i a ion o his ques ion is ha an a i ma i e answe would immedia ely gi e a p oo o he ine ia conjec u e. Obse e also ha he analogous ques ion o ee ac o s is ue, namely he in e sec ion o a ee ac o Ro Fnwi h any subg oup H≤Fnis a ee ac o o H. No e ha one o he esul s o Be gman in [1] is ha an a bi a y in e sec ion o e ac s o Fnis again a e ac o Fn.2 Conjec u e 8.3 (Equalize conjec u e).I Sis a se o homomo phisms om Fm o Fn, one o hem being injec i e, hen (Eq(S)) ≤m. Equi alen ly, Eq(S)is ine in Fm. Discussion. By es ic ing he homomo phisms in S o a gi en subg oup K≤ Fm, we see ha he wo s a emen s o he conjec u e a e in ac equi alen . By a esul o Golds ein-Tu ne [26], Eq(S) is ini ely gene a ed when Sis ini e. And by a esul o Be gman [1], i he e is a homomo phism ω:Fn→Fm such ha , o e e y φ∈S,φω is he iden i y o Fm hen Eq(S) is a ee ac o o Fm. In pa icula , (Eq(S)) ≤min his case. Using an injec i e elemen in S, we can es a e he conjec u e in he ollowing way: “Le Fm≤Fnand le Sbe an a bi a y se o mo phisms om Fm o Fn. Then, (Fix S)≤m(equi alen ly, Fix Sis ine in Fm)”. Wi h his new o mula- ion, he esul in [14] says ha he conjec u e is ue when he homomo phisms in Sall ex end o injec i e endomo phisms o Fn.2 Conjec u e 8.4 (Comp essed-ine conjec u e).E e y comp essed subg oup o Fnis ine . Discussion. Fi s , le us no e ha , using he Sch eie o mula, he conjec u e is clea ly ue o subg oups o Fno ini e index. 22 E. VENTURA Since e e y e ac o Fnis comp essed, bu he e a e comp essed subg oups which a e no e ac s, his conjec u e is s ic ly s onge han he ine ia con- jec u e. So a posi i e solu ion would imply ha endo- ixed subg oups o Fna e ine . Le Fmbe a comp essed subg oup o Fn, and le Sbe an a bi a y amily o homomo phisms om Fm o Fn. In [52], he ollowing is p o ed: i i is ue ha comp essed subg oups a e ine , hen Eq(S) is ine in Fm. So, his conjec u e seems o be o a simila le el o di icul y as he equalize conjec u e. 2 Ques ion 8.5.Is he e an algo i hm o decide i a gi en subg oup H≤Fnis ine ? Discussion. In [53] a simple g aphic algo i hm is gi en o decide i a gi en subg oup H≤Fnis comp essed. Howe e , he algo i hm seems o be unable o con ol subg oups no con aining H.2 Ques ion 8.6.Is he e an algo i hm o decide whe he a gi en subg oup H≤ Fnis 1-au o- ixed and, in his case, ind an au omo phism ixing H? Analogous ques ions can be asked o 1-mono- ixed, 1-endo- ixed, au o- ixed, mono- ixed and endo- ixed subg oups. Discussion. This ques ion asks o an algo i hm dual o he one p o ided by M. Lus ig in [35]. The desc ip ion o 1-au o- ixed subg oups p o ided by Theo em 5.8, al hough being qui e explici , does no seem o be good enough om he algo i hmic poin o iew. 2 Ques ion 8.7.Is he e any algeb aic cha ac e iza ion o 1-au o- ixed subg oups o Fn? Simila ly o he o he i e ypes o ixed subg oups. Discussion. Collins-Tu ne ga e in [9] such a cha ac e iza ion o he maximal ank case. The only esul ha is known in his di ec ion o he gene al case is Theo em 5.8 due o A. Ma ino and E. Ven u a. I is known ha e e y endo- ixed subg oup His pu e (i.e. x ∈Himplies x∈H) and ine . Howe e , hese condi ions, al hough being qui e es ic i e, especially he second one, a e no enough o cha ac e ize ixed subg oups. 2 Ques ion 8.8.Is he e any algo i hm o compu e he au o- ixed closu e Hco a gi en subg oup H≤Fn? Discussion. Gi en an au omo phism φo Fn, his ques ion amoun s o unde - s anding which elemen s o Fna e o ced o be ixed by φ, i some o he s a e. This can happen e en wi h independen elemen s (see he example in [38] men ioned in he p e ious sec ion). I is no clea how hese ela ionships wo k, no i is unde s ood how a subg oup Ho Fnde e mines i s au o- ixed closu e Hc.2 Conjec u e 8.9 (Ma ino-Ven u a, [38]).E e y au o- ixed subg oup o Fnis 1-au o- ixed. Discussion. This was conjec u ed in [38], which gi es much o he in o ma ion known in suppo o his conjec u e. The e, i was p o ed ha e e y au o- ixed ( esp. mono- ixed, endo- ixed) subg oup o Fnis a ee ac o o some 1-au o- ixed ( esp. 1-mono- ixed, 1-endo- ixed) subg oup o Fn. Howe e , an example was shown o a ee ac o o a 1-au o- ixed subg oup o Fnwi h n≥3, which is no e en endo- ixed. FIXED SUBGROUPS IN FREE GROUPS: A SURVEY 23 In pa icula , he conjec u e is ue o F2, and o maximum- ank au o- ixed subg oups. The same can conjec u ed wi h he p e ix mono- o endo- ins ead o au o-. 2 Ques ion 8.10.Wha a e he ela ionships (wi h espec o inclusion) be ween he amilies o 1-au o- ixed, 1-mono- ixed, 1-endo- ixed, au o- ixed, mono- ixed and endo- ixed subg oups o Fn? Discussion. I is easy o see ha hese six amilies o subg oups o Fncoincide when n= 2. By Theo em 6.5, hey also coincide in he maximal ank case. In gene al, apa om he ob ious inclusions be ween hem, he only in o ma- ion known is ha , o n≥3, he amily o 1-endo- ixed subg oups s ic ly con ains ha o 1-au o- ixed ones (see [40]). In be ween, he e is he amily o 1-mono- ixed subg oups, which is no known o coincide wi h any o he p e ious amilies. Conjec u e 8.9 and he discussion he e a e closely ela ed wi h his ques ion. 2 Ques ion 8.11.Le Fbe a ee g oup and φ∈End(F). Is Fix φine in F? Discussion. I Fis ini ely gene a ed, his ques ion has al eady been conside ed, and coincides wi h he ine ia conjec u e abo e (see 8.1). The only new ques ion asked he e is abou he in ini ely gene a ed case. The gene al Sco conjec u e is acuous in his case, since he inequali y (H)≤ (F) is immedia e o e e y subg oup H≤F. Howe e , he ine ia p ope y changes he si ua ion he e. I is no a i ial ques ion o ask i he ixed subg oup o an endomo phism o Fis ine in F. A simila ques ion abou equalize s can also be asked. No hing is known in his di ec ion. 2 Acknowledgmen s The au ho is indeb ed o he Ma hema ics Depa men o he Ci y College o New Yo k o he hospi ali y ecei ed du ing he academic yea 2000-2001, when his pape has been w i en. He is also indeb ed o J. Bu illo o se e al sugges- ions imp o ing he pape . The au ho g a e ully acknowledges pa ial suppo by he DGI (Spain) h ough g an BFM2000-0354 and by he DGR (Gene ali a de Ca alunya) h ough g an 2001BEA1400176. Re e ences [1] G.M. Be gman, Suppo s o de i a ions, ee ac o iza ions and anks o ixed subg oups in ee g oups, T ans. Ame . Ma h. Soc.,351 (1999), 1531-1550. [2] M. Bes ina, M. Handel, T ain acks and au omo phisms o ee g oups, Ann. o Ma h.,135 (1992), 1-51. [3] M. Bes ina, M. Handel, T ain acks o su ace homeomo phisms, Topology,34(1) (1995), 109-140. [4] O. Bogopolski, Classi ica ion o au omo phisms o he ee g oup o ank 2 by anks o ixed- poin subg oups, J. G oup Theo y,3(2000), 339–351. [5] A.J. Casson, S.A. 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