A short proof of a theorem of Brodski

A short proof, using graphs and groupoids, is given of Brodski˘i's theorem that torsion-free one-relator groups are locally indicable.


Introduction
In 1980, Sergei Brodskiȋ 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 Brodskiȋ'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 Brodskiȋ'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.
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.

Preliminaries
A graph 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 → V is defined by t(e) = i(e −1 ).A path in Γ from the vertex u to the vertex v is a sequence e 1 , . . ., e n of edges, with i(e 1 ) = u, i(e j ) = t(e j−1 ) for 2 ≤ j ≤ n, and t(e n ) = v. (We also call u the initial vertex, and v the terminal vertex of P .)The path P is reduced if e j = e −1 j−1 for all 2 ≤ j ≤ n, and closed if u = v.A reduced closed path is cyclically reduced if, in addition, e n = 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 v forms a group π(Γ, v), called the path group, or fundamental group, of Γ (based at v).It is equal to the vertex group at v of the groupoid F (Γ).It is a free group, and every free group arises in this way.
A presentation Γ | R of a groupoid G consists of: 1) a graph Γ; and 2) a set R of 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 R and E = E(Γ) which are compatible in the sense that, if α, β ∈ R with 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 G is indicable if it admits an infinite cyclic homomorphic image.It is locally indicable if every non-trivial, finitely generated subgroup is indicable.

The main result Theorem 3.1. Let G be a groupoid given by a staggered presentation Γ | R in which no element of R is a proper power. Then every vertex group of G is locally indicable.
Brodskiȋ's theorem is the special case of Theorem 3.1 in which V (Γ) and R are 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 | r m be a one-relator group, where m ≥ 2 and r is not a proper power.Let Ḡ = X | r be the corresponding torsion-free one-relator group.Then there is a short exact sequence in which F is a free product of cyclic groups [4].If H is a torsion-free subgroup of G, then H ∩ F is free, so locally indicable.Hence H is 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 G v of G at v. Suppose H is generated by reduced closed paths γ 1 , . . ., γ n at v. Since H is non-indicable, it has finite abelianisation, and so there are n words W 1 , . . ., W n in the free group on n generators x 1 , . . ., x n , such that: 1) the abstract group x 1 , . . ., x n | W 1 , . . ., W n has finite abelianisation; and 2) each path W j (γ 1 , . . ., γ n ) (1 ≤ j ≤ n) belongs to N (R).Because of 2) there is an identity: for each j, where each α j,k is an element of R or its inverse, and each δ j,k is a path in Γ from v to the initial (and terminal) vertex of α j,k .We will refer to the collection of paths γ j , words W j and identities (1) as a datum, ∆ say.There is nothing in the definition of a datum which enforces the nontriviality of the subgroup H generated by the γ j , 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 H vanishes.We will do this by induction on L(∆) − M (∆), where L(∆) is the sum of the lengths of all the paths δ j,k and α j,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 Γ 0 containing all the paths δ j,k and α j,k (and hence all the γ j ), and R by the subset R 0 = {α j,k | 1 ≤ j ≤ n, 1 ≤ k ≤ m(j)}.Note that Γ 0 is a finite graph, and R 0 is a finite set.This gives a new (finite) presentation Γ 0 | R 0 of a groupoid G 0 ; the paths γ j generate a subgroup H 0 of G 0 ; the inclusion of Γ 0 in Γ induces a natural homomorphism G 0 → G which maps H 0 onto H; and the presentation of G 0 is staggered under the restriction of the orders on E(Γ) and R to E(Γ 0 ) and R 0 respectively.In particular, if we prove that H 0 = {1}, then it follows that H = {1}, as desired.
From now on, we assume that G = G 0 , etc.
Choose an epimorphism G v → Z of groups and extend it to an epimorphism θ : G → Z of groupoids.Corresponding to θ we construct infinite cyclic coverings Γ of Γ and G of G as 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 P in Γ, beginning at a vertex v, say, and any integer n, there is a unique path P n in Γ , beginning at (v, n), with π(P n ) = P .(We call P n the lift of P beginning at (v, n).)If P ends at a vertex u, then P n ends at (u, n + θ(P )).In particular, for each r ∈ R, each r n is a closed path, since r is closed and θ(r) = 0. Let R be the set {r n | r ∈ R, n ∈ Z} of closed paths in Γ , and define G to be the groupoid Γ | R .
Since H is non-indicable, we must have θ(H) = 0. Hence each path γ j lifts to a closed path γ j at v := (v, 0).If δ j,k is the lift of δ j,k that begins at v , and α j,k is the lift of α j,k that begins at the terminal vertex of δ j,k , then we have identities for each 1 ≤ j ≤ n.
We introduce linear orderings on E(Γ ) and R by: 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 , W j and identities (2).Then H is non-indicable, by the identities (2), and H is mapped onto H by π.We show that the inductive hypothesis applies to H .It follows that H , and hence H, vanishes.
Case 2: Now assume that G v is 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. If R = ∅, then G = F (Γ) is a free groupoid, so G v is a free group.But G v is also non-indicable, so G v = {1} and hence H = {1}.
If R = {r} is a singleton set, then G v is a one-relator group.Since G v is 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 r is cyclically reduced and not a proper power, r is this cycle (traversed in one of the two possible directions), so again H = G v = {1}.Moreover, note that each edge in r occurs precisely once in r.
For the general case, we take the slightly stronger property noted above to be the inductive hypothesis: namely that G v = {1} and every edge occurring in any relation r ∈ R occurs precisely once in r.Now suppose that r max is the greatest relation in R (with respect to the given linear ordering).Let e = max(r max ).Suppose first that Γ = Γ \ {e} is connected.Then G = Γ | R \ {r max } 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(r min ) occurs precisely once in r min , where r min is the least relator in R \{r max } (and hence in R).Thus G 2 = Γ\{f } | R\{r min } is isomorphic to G, so has nonindicable vertex groups.By inductive hypothesis the vertex groups of G 2 are 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 Γ 3 and Γ 4 , say.For each relator other than r max must be a path in one of Γ 3 , Γ 4 , so R \ {r max } splits as a disjoint union R 3 ∪ R 4 , and we have two groupoids G 3 = Γ 3 | R 3 and G 4 = Γ 4 | R 4 .Since G has nonindicable vertex groups, so does at least one of G 3 , G 4 (say G 3 ).Now R 3 cannot be empty, for then Γ 3 would be a tree, and no cyclically reduced closed path could contain e = max(r max ), a contradiction.Now apply the same argument as above, taking r min to be the least relator in R 3 .
This completes the proof.