A Short proof of a theorem of Brodskii

Publicacions Matem`atiques, Vol. 44 (2000), 641–647 A SHORT PROOF OF A THEOREM OF BRODSKI˘ I James Howie Abstract A short proof, using graphs and groupoids, is given of Brodski˘ı’s theorem that torsion-free one-relator groups are locally indicable. 1. Introduction In 1980, Sergei Brodski˘ı announced [2] the result, previously conjectured by Gilbert Baumslag [1], that every torsion-free one-relator group is locally indicable, that is, every nontrivial, finitely generated subgroup has an infinite cyclic homomorphic image. His algebraic proof was published in full in 1984 [3]. Around the same time, I independently obtained Brodski˘ı’s theorem, and published a slightly more general version in [7], with a topological proof: a one-relator quotient of a free product of locally indicable groups is locally indicable, provided the relator is neither a proper power nor conjugate to an element of one of the free factors. A further version of the theorem was later proved by John Hempel [5]: the quotient of a surface group by a single relator that is not a proper power is locally indicable. This paper arose as a response to requests from colleagues —notably Warren Dicks— for a proof of Brodski˘ı’s theorem more accessible than those in [3], [7]. In particular the topology used in [7] seemed to cause some difficulty. Here I present a straightforward proof of the theorem, using groupoids. It is essentially my proof from [7], restricted to the original case of a torsion-free one-relator group, with as much of the topology as possible translated into algebra. The only remaining topology is the notion of an infinite cyclic cover of a graph or groupoid. For more detailed background material on graphs and groupoids, the best reference is [6], but for completeness I have included some elementary definitions in §2 below, and a description of the construction of infinite cyclic covers in §3. 642 J. Howie Acknowledgement. I first presented a version of this proof in a seminar to mark the retirement of my PhD supervisor, Philip Higgins. I am grateful to him for his support. I am also grateful to Warren Dicks for encouraging me to publish this material, and for pointing out Corollary 3.2. 2. Preliminaries Agraph Γ consists of a set V=V(Γ) of vertices and a set E=E(Γ) of edges, together with a map i:E→V(the initial vertex map), and a fixed-point-free involution e→ e−1:E→E. The terminal vertex map t:E→Vis defined by t(e)=i(e−1). A path in Γ from the vertex uto the vertex vis a sequence e1,...,e nof edges, with i(e1)=u, i(ej)=t(ej−1) for 2 ≤j≤n, and t(en)=v. (We also call uthe initial vertex, and vthe terminal vertex of P.) The path Pis reduced if ej=e−1 j−1for all 2 ≤j≤n, and closed if u=v. A reduced closed path is cyclically reduced if, in addition, en=e−1 1. A closed path is a proper power if it is obtained by repeating a closed path two or more times. A graph is connected if any two vertices are joined by a path. The set of all reduced paths forms a groupoid F(Γ) under juxtaposition (followed by cancellation of any resulting inverse pairs of consecutive edges), called the free groupoid on Γ (see [6] for details). The set of all reduced closed paths at a vertex vforms a group π(Γ,v), called the path group,orfundamental group, of Γ (based at v). It is equal to the vertex group at vof the groupoid F(Γ). It is a free group, and every free group arises in this way. A presentation Γ|Rof a groupoid Gconsists of: 1) a graph Γ; and 2) a set Rof cyclically reduced closed paths in Γ, such that G=F(Γ)/N (R), where N(R) denotes the smallest normal subgroupoid containing R. The presentation is staggered if there are linear orderings on the sets Rand E=E(Γ) which are compatible in the sense that, if α, β ∈Rwith α<β, then max(α)<max(β) and min(α)<min(β), where max and min denote the greatest and least edges occurring in a path (under the given linear ordering on E). A group Gis indicable if it admits an infinite cyclic homomorphic image. It is locally indicable if every non-trivial, finitely generated subgroup is indicable. A Short Proof of A Theorem of Brodski˘ ı 643 3. The main result Theorem 3.1. Let Gbe a groupoid given by a staggered presentation Γ|Rin which no element of Ris a proper power. Then every vertex group of Gis locally indicable. Brodski˘ı’s theorem is the special case of Theorem 3.1 in which V(Γ) and Rare singleton sets (and E(Γ) has an arbitrary ordering). Corollary 3.2. Any torsion-free subgroup of a one-relator group is locally indicable. Proof: Let G=X|rmbe a one-relator group, where m≥2 and ris not a proper power. Let ¯ G=X|rbe the corresponding torsion-free one-relator group. Then there is a short exact sequence 1→F→G→¯ G→1 in which Fis a free product of cyclic groups [4]. If His a torsion-free subgroup of G, then H∩Fis free, so locally indicable. Hence His an extension of a locally indicable group by a locally indicable group, so is locally indicable. Proof of Theorem 3.1: Suppose the theorem were false. Then for some G=Γ|Ras in the theorem, and some vertex v∈V(Γ), there would be a finitely generated, non-indicable subgroup H={1}of the vertex group Gvof Gat v. Suppose His generated by reduced closed paths γ1,...,γ nat v. Since His non-indicable, it has finite abelianisation, and so there are nwords W1,...,W nin the free group on n generators x1,...,x n, such that: 1) the abstract group x1,...,x n|W1,...,W nhas finite abelianisation; and 2) each path Wj(γ1,...,γ n)(1≤j≤n) belongs to N(R). Because of 2) there is an identity: Wj(γ1,...,γ n)=(δj,1αj,1δ−1 j,1)···(δj,m(j)αj,m(j)δ−1 j,m(j))(1) for each j, where each αj,k is an element of Ror its inverse, and each δj,k is a path in Γ from vto the initial (and terminal) vertex of αj,k. We will refer to the collection of paths γj, words Wjand identities (1) as a datum, ∆ say. There is nothing in the definition of a datum which enforces the nontriviality of the subgroup Hgenerated by the γj, so data exist for the trivial subgroup also. In fact, we will prove the theorem by showing that, for any datum as above, the corresponding subgroup Hvanishes. We will do this by induction on L(∆) −M(∆), where L(∆) is the sum 644 J. Howie of the lengths of all the paths δj,k and αj,k, and M(∆) is the number of distinct vertices visited by these paths. Clearly M(∆) ≤L(∆), so induction on L(∆) −M(∆) makes sense. The first step is to replace Γ by the smallest subgraph Γ0containing all the paths δj,k and αj,k (and hence all the γj), and Rby the subset R0= {αj,k |1≤j≤n, 1≤k≤m(j)}. Note that Γ0is a finite graph, and R0is a finite set. This gives a new (finite) presentation Γ0|R0of a groupoid G0; the paths γjgenerate a subgroup H0of G0; the inclusion of Γ0in Γ induces a natural homomorphism G0→Gwhich maps H0 onto H; and the presentation of G0is staggered under the restriction of the orders on E(Γ) and Rto E(Γ0) and R0respectively. In particular, if we prove that H0={1}, then it follows that H={1}, as desired. From now on, we assume that G=G0, etc. Case 1: Assume that the vertex group Gvof Gat vis indicable. Choose an epimorphism Gv→Zof groups and extend it to an epimorphism θ:G→Zof groupoids. Corresponding to θwe construct infinite cyclic coverings Γof Γ and Gof Gas follows. Firstly we define V(Γ):=V(Γ) ×Z;E(Γ):=E(Γ) ×Z;i(e, n):=(i(e),n) and (e, n)−1:= (e−1,n+θ(e)) to get a graph Γ. The projections onto the first coordinates determine a graph homomorphism π:Γ →Γ, called a covering projection. This satisfies the (easily verified) path lifting property: given any path Pin Γ, beginning at a vertex v, say, and any integer n, there is a unique path Pnin Γ, beginning at (v,n), with π(Pn)=P. (We call Pnthe lift of Pbeginning at (v,n).) If Pends at a vertex u, then Pnends at (u, n +θ(P)). In particular, for each r∈R, each rnis a closed path, since ris closed and θ(r) = 0. Let Rbe the set {rn|r∈R, n ∈Z}of closed paths in Γ, and define Gto be the groupoid Γ|R. Since His non-indicable, we must have θ(H) = 0. Hence each path γj lifts to a closed path γ jat v:= (v,0). If δ j,k is the lift of δj,k that begins at v, and α j,k is the lift of αj,k that begins at the terminal vertex of δj,k, then we have identities Wj(γ 1,...,γ n)=(δ j,1α j,1(δ j,1)−1)···(δ j,m(j)α j,m(j)(δ j,m(j))−1)(2) for each 1 ≤j≤n. We introduce linear orderings on E(Γ) and Rby: (e, n)<(f,m)ife<f or if e=fand n<m; rn<s mif r<s or if r=sand n<m. A Short Proof of A Theorem of Brodski˘ ı 645 It is clear that these are compatible, and hence that Γ|Ris a staggered presentation. Let Hbe the subgroup of Ggenerated by the paths γ j(1 ≤j≤n), and let ∆be the datum consisting of the γ j,Wjand identities (2). Then His non-indicable, by the identities (2), and His mapped onto Hby π. We show that the inductive hypothesis applies to H. It follows that H, and hence H, vanishes. Clearly L(∆)=L(∆). Let Γ1be the smallest subgraph of Γcontaining all the paths δ j,k and α j,k. Then M(∆)=|V(Γ1)|and M(∆) = |V(Γ)|. Moreover, Γ1is mapped surjectively onto Γ by π, and it suffices to show that this surjection is proper on vertices. If not, then by construction the restriction of πto Γ1must be bijective both on edges and on vertices, so a graph isomorphism Γ1→Γ. But there is at least one closed path βinΓatvwith θ(β) = 1. Under the graph isomorphism π−1:Γ→Γ1,βis mapped onto the unique path βbeginning at v=(v,0) such that π(β)=β. But by definition βends at (v,θ(β)) = (v,1) =(v,0). Hence (v,0),(v,1) ∈V(Γ1) with π(v,0) = π(v,1) = v, contradicting the assumption that π:V(Γ1)→V(Γ) is injective. This contradiction completes the proof in Case 1. Case 2: Now assume that Gvis not indicable. Note that the argument in Case 1 shows that this must include the initial case of the induction. The proof in this case is a second induction, this time on the number of elements in the relation set R.IfR=∅, then G=F(Γ) is a free groupoid, so Gvis a free group. But Gvis also non-indicable, so Gv= {1}and hence H={1}. If R={r}is a singleton set, then Gvis a one-relator group. Since Gvis non-indicable, it must be finite cyclic, and so π(Γ,v) is cyclic. In other words, Γ has first Betti number 1, and so contains a single nontrivial cycle. Since ris cyclically reduced and not a proper power, ris this cycle (traversed in one of the two possible directions), so again H=Gv={1}. Moreover, note that each edge in roccurs precisely once in r. For the general case, we take the slightly stronger property noted above to be the inductive hypothesis: namely that Gv={1}and every edge occurring in any relation r∈Roccurs precisely once in r. 646 J. Howie Now suppose that rmax is the greatest relation in R(with respect to the given linear ordering). Let e= max(rmax). Suppose first that Γ = Γ\{e}is connected. Then G =Γ |R\{rmax} cannot have indicable vertex groups, for then so would G. By induction each vertex group of G is trivial, and each edge occurring in each relator occurs precisely once in that relator. In particular f= min(rmin) occurs precisely once in rmin, where rmin is the least relator in R\{rmax}(and hence in R). Thus G2=Γ\{f}|R\{rmin} is isomorphic to G, so has nonindicable vertex groups. By inductive hypothesis the vertex groups of G2are trivial, and each edge occurring in any of its relators occurs precisely once in that relator. Hence the same is true for G, and we are done. A similar argument works if we suppose that G has two components Γ3and Γ4, say. For each relator other than rmax must be a path in one of Γ3,Γ 4,soR\{rmax}splits as a disjoint union R3∪R4, and we have two groupoids G3=Γ3|R3and G4=Γ4|R4. Since G has nonindicable vertex groups, so does at least one of G3,G4(say G3). Now R3cannot be empty, for then Γ3would be a tree, and no cyclically reduced closed path could contain e= max(rmax), a contradiction. Now apply the same argument as above, taking rmin to be the least relator in R3. This completes the proof. References [1] G. Baumslag, Some problems on one-relator groups, in: “Proceedings of the Second International Conference on the Theory of Groups” (Australian Nat. Univ., Canberra, 1973), Lecture Notes in Math. 372, Springer, Berlin, 1974, pp. 75–81. [2] S. D. Brodski˘ ı, Equations over groups and groups with one defining relation, Uspekhi Mat. Nauk 35(4) (1980), 183; Russian Math. Surveys 35(4) (1980), 165. [3] S. D. Brodski˘ ı, Equations over groups and groups with one defining relation, Sibirsk. Mat. Zh. 25(2) (1984), 84–103; Siberian Math. J. 25(2) (1984), 235–251. [4] J. Fischer, A. Karrass and D. Solitar, On one-relator groups having elements of finite order, Proc. Amer. Math. Soc. 33 (1972), 297–301. [5] J. Hempel, One-relator surface groups, Math. Proc. Cambridge Philos. Soc. 108(3) (1990), 467–474. A Short Proof of A Theorem of Brodski˘ ı 647 [6] P. J. Higgins,“Notes on categories and groupoids”, Van Nostrand Reinhold Mathematical Studies 32, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. [7] J. Howie, On locally indicable groups, Math. Z. 180(4) (1982), 445–461. Department of Mathematics Heriot-Watt University Riccarton Edinburgh EH14 4AS United Kingdom E-mail address:[email protected] Primera versi´o rebuda el 27 de juny de 2000, darrera versi´o rebuda el 2 de setembre de 2000.