Coherence and one-relator products of locally indicable groups

We extend several results of Helfer, Wise, Louder and Wilton related to coherence in one-relator groups to the more general setting of one-relator products of locally indicable groups. The methods developed to do so also give rise to a new proof of a theorem of Brodsky.


Introduction
In a major recent breakthrough, Louder and Wilton [14] -and independently Wise [18] -have shown that one-relator groups with torsion are coherent. In other words, every finitely generated subgroup of such a group has a finite presentation. This gives a partial answer to an old question of G. Baumslag [1].
A sizeable body of work over the past 40 years, starting with the papers of Brodskiȋ [4,3] and the authors [9,17], has shown that much of one-relator group theory extends to one-relator products of locally indicable groups. (Recall that a group is locally indicable if each of its non-trivial finitely generated subgroups admits an epimorphism onto the infinite cyclic group Z.) In that spirit, we prove in the current paper the natural analogue of this coherence result, as follows.
Theorem A. Let G λ , λ ∈ Λ, be a collection of coherent, locally indicable groups, let S ∈ * λ G λ be a cyclically reduced word of length at least 2, and let n > 1 be an integer. Then the one-relator product and Wilton [13], most of which also applies to torsion-free one-relator groups and yields partial results in support of the idea that they too are coherent. We are also able to prove in the current article natural analogues of many of these results in the setting of one-relator products. We describe these generalizations below.
We are happy to acknowledge that in the construction of our proofs we have leant heavily on the arguments of Helfer, Wise, Louder and Wilton in the articles cited above, many of which can be readily transported into our framework. There are of course also some additional difficulties in the more general setting, but we have been able to resolve these.
A two-dimensional CW-complex Y is said to have non-positive immersions if, for every compact, connected, non-contractible 2-complex Y admitting an immersion Y → Y , the Euler characteristic χ(Y ) is non-positive.
Motivated by Baumslag's conjecture, Helfer and Wise [8] and independently Louder and Wilton [13] show that torsion-free one-relator group presentations have the non-positive immersions property, and use this fact in different applications. For example, Louder and Wilton [13] show that non-trivial finitely generated subgroups of torsion-free one-relator groups have finitely generated Schur multiplier -indeed the rank of the multiplier is strictly less than the rank of the abelianisation. As a consequence, one can rule out many incoherent groups such as Thompson's group F , the wreath product of Z with Z, and the direct product of two non-abelian free groups, as subgroups of torsion-free one-relator groups.
In the present note, we prove relative analogues of some of these results for one-relator products of locally indicable groups. (The results also follow for staggered products of locally indicable groups, since these can be constructed as iterative one-relator products.) Now it is easy to show that each component of a 2-complex with non-positive immersions has locally indicable fundamental group, so the local indicability criterion can be omitted from the statement of the first of these results.
The following construction occurs throughout the paper, so it is convenient to give it a name. Following [8], we say that a CW-complex Y is a simple enlargement of a CW-complex X if Y is obtained from X by adjoining a 1-cell e and at most one 2-cell α, and in the latter case the attaching path R for α: (1) is a closed combinatorial edge-path in X (1) ∪ e and involves e; (2) is not freely homotopic in X ∪ e to a path that crosses e fewer times; and (3) does not represent a proper power in π 1 (X ∪ e). In practice, we will always assume that Y is connected. So X has either one or two components -say X 1 , X 2 in the latter case. Then π 1 (X ∪ e) is isomorphic to a free product π 1 (X) * Z or π 1 (X 1 ) * π 1 (X 2 ), while if Y \ X has a 2-cell α with attaching map in the homotopy class of R ∈ π 1 (X ∪ e) then π 1 (Y ) is a one-relator product π 1 (X) * Z R or π 1 (X 1 ) * π 1 (X 2 ) R .
Theorem B. Let X be a 2-complex with non-positive immersions, and let Y be a simple enlargement of X. Then Y has non-positive immersions.
Applying this to the case where X is one-dimensional, we recover the main result of Helfer and Wise [8,Theorem 1.3] and of Louder and Wilton [13,Corollary 4]: Corollary 1.1. Every torsion-free one-relator group presentation has non-positive immersions.
The proofs of Theorems A and B follow a similar pattern to those of Wise in [18] and of Helfer and Wise in [8], respectively. They avoid the explicit use of towers, pictures, Magnus induction, or similar tricks from one-relator theory. Instead they use only known facts about onerelator products of locally indicable groups (which of course do use the said tricks in their own proofs).
We also prove an analogue of the theorem of Louder and Wilton about the second Betti number β 2 (K) of a finitely generated subgroup K of a torsion-free one-relator group.
Let us say that a group G has the second Betti number property if, for any non-trivial finitely generated subgroup K of G, the second Betti number β 2 (K) of K is strictly less than the first Betti number β 1 (K). Louder and Wilton [13,Corollary 5] show that torsion-free one-relator groups have the second Betti number property. Below we prove an analogous result for one-relator products.
Theorem C. Let G := ( * λ∈Λ G λ ) R be a one-relator product of locally indicable groups, each with the second Betti number property, where R ∈ * λ G λ is cyclically reduced of length at least 2, and not a proper power . Then G has the second Betti number property.
Indeed, we prove a slight generalisation of this theorem as follows.
Note that G has the second Betti number property if and only if the zero function is a second Betti bounding function, and so Theorem C follows from : Theorem D. Let X be a 2-complex with one or two components, each having a locally indicable fundamental group, and let Y be a simple enlargement of X. If F : N → N is a second Betti bounding function for the fundamental groups of the components of X, then F is a second Betti bounding function for π 1 (Y ).

Example
Let Λ be a limit group. Then Λ can be constructed from a collection of free abelian groups of finite rank by a series of constructions which are either free products, free products with cyclic amalgamation, or HNN extensions with cyclic amalgamation. These constructions are special cases of simple enlargements. There is a finite upper bound r > 1 to the rank of abelian subgroups of Λ. The map n → n(r−1)(r−2) 2 is a second Betti bound for Z r , and hence also for Λ, by iterated applications of Theorem D.
In particular, if Λ is hyperbolic (or more generally if Λ has no abelian subgroup of rank greater than 2), then the zero function is a second Betti bound for Λ, so Λ has the second Betti number property.
We also consider the existence or otherwise of certain incoherent subgroups and prove that their non-existence is preserved under simple extensions.
R be a one-relator product of locally indicable groups. Let K < G be a subgroup of G isomorphic to one of the following: (1) a free abelian group of rank r > 2; (2) the wreath product Z wr Z; (3) Thompson's group F ; (4) the direct product of two free groups of rank at least 2; (5) a right angled Artin group whose underlying graph is connected, with more than one vertex, and without cut points. Then there exists a unique λ ∈ Λ and a unique right coset G λ g of G λ in G such that K < G g λ . Remarks. In the last part of Theorem E, all the conditions on the graph of the right angled Artin group are necessary. For a lone vertex, K is just an infinite cyclic group, which could belong to several or none of the G g λ . For a connected graph with a cut vertex, K is a free product of two right angled Artin groups K 1 , K 2 with cyclic amalgamation, so can be expressed as a one-relator product of the locally indicable groups K 1 , K 2 . If the graph has a separating edge then K can again be expressed as a one-relator product of two right angled Artin groups, where the relator is a commutator. Finally, if the graph is disconnected, then K is a free product, and it is easy to construct counterexamples in this case.
Note also that parts 1 and 4 of Theorem E are covered by part 5, in the case of complete graphs and complete bipartite graphs respectively. We have included them as separate parts of the theorem because there are more straightforward proofs in those cases. Moreover part 4 of Theorem E is of particular interest. It complements a result of Brodskiȋ [4,Theorem 8] which gives strong restrictions on subgroups of G that decompose as direct products with at least one non-free direct factor.
The proof of Theorem E makes use of the following result of Brodskiȋ [4,Theorem 6].
Theorem F. Let G λ , λ ∈ Λ, be a collection of locally indicable groups, let R ∈ * λ G λ be a cyclically reduced word of length at least 2, and let If g ∈ G and λ, µ ∈ Λ are such that the intersection in G of G λ and g −1 G µ g is not cyclic, then µ = λ and g ∈ G λ .
It turns out that our methods also yield a new proof of this important result. Since [4] is not to our knowledge available online or in translation, we feel that is worthwhile including our proof in this article as well.
The remainder of the paper is organised as follows. In §2 we recall some relevant definitions and previous results. In §3 we prove Theorem A and note stronger versions of Theorems E and F which hold when the relator is a proper power. In §4 we prove Theorem C and D. In §5 we prove the first three parts of Theorem E. In §6 we prove Theorem F. Finally in §7 and §8 we prove the last two parts of Theorem E.

Some Technical results we shall need
A fundamental result about one relator products of locally indicable groups is the Freiheitssatz, due independently to the authors and S. Brodskiȋ. We shall need this result frequently. [11], [17]). Let A, B be locally indicable groups and R ∈ A * B a cyclically reduced word of length at least two.
The natural map A → A * B R is injective.
When proving Theorems F and E, we shall need the following decomposition of the cohomology of a one relator product: * λ∈Λ G λ R n be a one-relator product of locally indicable groups G λ , where R is cyclically reduced of length at least 2 and not a proper power in * λ G λ and n ≥ 1. Let C be the cyclic subgroup of G generated by R, and let M be a ZG-module. Then the restriction maps are isomorphisms for k > 2 and an epimorphism for k = 2.
Proof. Let K := G µ 1 ∩ G g µ 2 , let M be a ZK-module, and let k ≥ 2 be an integer. By Shapiro's Lemma we have Thus by Theorem 2.2 we have an epimorphism , where g ranges across double-coset representatives for CgK, G λ gK respectively. But for the two terms on the right hand side of this equation corresponding to G µ 1 and G g µ 2 , we have K = K ∩ G µ 1 and is also an epimorphism, and it follows that H k (K; M ) = 0.
A map between CW-complexes is said to be combinatorial if it maps the interior of each cell homeomorphically onto the interior of a cell (of the same dimension). An immersion of CW-complexes is a combinatorial map which is locally injective. For example, every covering space and every inclusion of subcomplexes is an immersion, as is the composite of any finite chain of immersions. In particular a tower (a composite of inclusion maps and covering maps) is an immersion.
We make frequent use of the following useful fact.
The factorisation of φ in Lemma 2.4 is not known to be unique. The collection of all such factorisations forms a category in the obvious way. There are two approaches to proving the lemma. Louder and Wilton [13, Lemma 4.1] perform a higher-dimensional version of Stallings folding techniques, which constructs an initial object in the category. The alternative approach -see [11,Lemma 3.1] or [18,Lemma 2.2] is to take φ to be a maximal tower lift of φ, which can be shown to yield a terminal object in the category of factorisations. So the two constructions satisfy universal and co-universal properties respectively. For our purposes we require only the existence of the factorisation; the precise construction used plays no role.
Following Louder and Wilton [14], if Y = X ∪ e ∪ α is a simple enlargement of a 2-complex X, we define a map f : Y → Y of 2complexes to be a branch map if it is combinatorial on the complement of f −1 (α); locally injective in the complement of the preimage f −1 (ξ) of a single point ξ in the interior of α; and if each 2-cell α in f −1 (α) maps to α as a cyclic branched cover branched over ξ. We will refer to the degree n ∈ Z + of this branched cover as the branch index of α .
The following is a natural example of a branch map to Y which forms a key tool in the study of one-relator products in which the relator is a proper power. For a given positive integer n, replace α by a 2-cell α n whose attaching path is the n'th power of that of α. Let Y n denote the resulting complex, and define ψ n : Y n → Y to be the identity on the complement of α n , and on α n the n-fold cyclic cover to α, branched over ξ. Then ψ n : Y n → Y is clearly a branch map. We call it the n-fold branched cover of Y .
In this paper we will make use of van Kampen diagrams and also their duals, which are known as pictures. Pictures were introduced by Rourke in [15] and adapted to the relative context by the second author [17]. A picture arises from a continuous map Σ → X, where Σ is a compact orientable surface and X a 2-complex, using transversality. It consists of a finite collection of discs or (fat) vertices, whose interiors each map homeomorphically onto the interior of a 2-cell of X, and a properly embedded 1-submanifold of the complement of the interiors of the discs, each component of which is called an arc, carries a transverse orientation and is labelled by a 1-cell of X. A small regular neighbourhood of each arc is mapped to the corresponding 1-cell in the direction of the transverse orientation.
If Y = X ∪ e ∪ α is a simple enlargement, then any picture over Y can be made into a relative picture by removing all discs that do not map to α, and all arcs that do not map to e.
For more details on pictures, and an example of their usefulness in group theory we refer the reader to [7], [12].

The Torsion Case
Let X be a 2-complex such that every connected component of X has locally indicable fundamental group. Let Y := X ∪ e ∪ α be a simple enlargement of X, and let R denote the closed combinatorial path in X (1) ∪ e along which α is attached.
From the definition of simple enlargement, the path R does not represent a proper power in π 1 (X ∪ e). For each positive integer n, let ψ n : Y n → Y be the n-fold branched cover of Y , as defined in §2. In particular Y n = X ∪ e ∪ α n where α n has attaching map R n .
We subdivide e at its midpoint x, forming two half-edges, and choose x as the basepoint for Y . We orient these half-edges so that x is the initial point of each.
Weinbaum's Theorem [9, Corollary 3.4] tells us that no proper closed cyclic subpath of R represents the identity element of G := π 1 (Y ). The L points where R meets x split it into L closed subpaths r 1 , . . . , r L such that R = r 1 r 2 · · · r L . If we let g j := r 1 · · · r j , j = 1, . . . , L denote the initial segments of R, then these represent pairwise distinct elements of G. Choose i, j ∈ {1, . . . , L} to minimise g −1 i g j (with respect to the chosen left ordering on G). The indices i and j are unique with respect to this property, as the following argument shows. Let k ∈ {1, . . . , L}. Then by choice of i, j we have Left multiplying this inequality by g i gives g k ≥ g j . It follows that g j is the unique minimal element of {g 1 , . . . , g L }, so the index j is unique. Hence if k, ∈ {1, . . . , L} are such that g −1 g k = g −1 i g j , we have k = j and hence g = g i , and so also = i. The index i is therefore also unique.
Replacing R by its cyclic conjugate g −1 j Rg j , we can write R as U · V where U −1 = V = g −1 i g j in G, so that U −1 is the unique minimal cyclic subword of R −1 (with respect to ≤), and V is the unique minimal cyclic subword of R (with respect to ≤). In particular, neither U nor V is a piece of R, in the sense of small cancellation theory. Moreover, every proper initial segment of U · V is equal in G to g −1 j g k for some k = j, and hence is positive in the left ordering of G, by the minimality of g j .
Similarly, every proper terminal segment of U · V is inverse in G to a proper initial segment, and so is negative in the left ordering.
Let us assume that R already has the form U · V as above. We refer to the edge of ∂α whose midpoint is the starting point of R as the associated 1-cell of α.
Now suppose that f : Y → Y is a branch map. In particular f is an immersion on the complement of the preimage of a single point in the interior of α. Suppose also that α ∈ f −1 (α) is a 2-cell with branch index n. Then there are n choices of attaching path for α that are mapped by f to the path (U V ) n . Each of these paths starts at the midpoint of an edge in f −1 (e). We refer to these edges as the low edges of α . Similarly, there are n choices of attaching path that are mapped to (V U ) n ; each starts at the midpoint of an edge called a high edge of α .
For each cell α ∈ f −1 (α) we choose one of the low edges of α and call it the associated 1-cell of α . The attaching path for α starting at the midpoint of the associated 1-cell is called the distinguished attaching path for α . Note that, since f is an immersion on a neighbourhood of the 1-skeleton of Y , no edge can be a low edge (resp. high edge) of more than one 2-cell in f −1 (α). In particular, distinct 2-cells in f −1 (α) have distinct associated 1-cells.
If the low (resp. high) edge e of a 2-cell α ∈ f −1 (α) is a free edge, then we call the resulting collapse Y → Y {e , α } a low-edge collapse (resp. a high-edge collapse).
The following slight generalisation of the notion of low-(resp. high-) edge collapse will also be useful. Suppose that the attaching path of α is a k'th power S k , where the low edge e of α appears precisely once in S and does not appear in the attaching path of any 2-cell other than α . Let Y := Y {α , e }. Then we say that e is an almost-free edge of α , and that replacing Y by Y is a low-(resp. high-) edge almost collapse. Note that in this case π 1 (Y ) ∼ = π 1 (Y ) * C, where C is cyclic of order k. Note also that we include low-and high-edge collapses in this definition; these correspond to the case where k = 1.
Theorem 3.1. Let X be a 2-complex such that every connected component of X has locally indicable fundamental group, and let Y be a simple enlargement of X. Suppose that f : Y → Y is a branch map, with Y compact and connected, and let A := Y f −1 (X). Let Z be the subcomplex obtained from Y by removing all the 2cells in A and their associated 1-cells. Suppose also that, for some Then Y can be transformed to T through a sequence of low-edge almost-collapses.
Proof. If Y = X ∪e then T = Z = Y and there is nothing to prove. So for the rest of the proof we consider only the case where Y = X ∪ e ∪ α.
Let T denote the collection of subcomplexes of Y that transform to T through a sequence of low-edge almost-collapses. Then T ∈ T so T is non-empty. Clearly T is partially ordered via inclusion. Since f −1 (α) is finite, it follows that T must have a maximal element T , say. The assertion of the theorem is that T = Y , so we argue by contradiction, beginning from the assumption that T = Y . Note also that π 1 (T ) is a free product of π 1 (T ) together with a finite number of finite cyclic groups. Since f * (π 1 (T )) = {1} and G is locally indicable, it follows that f * (π 1 (T )) = {1} in G.
Consider the subset A of A consisting of those 2-cells not in T whose associated 1-cells meet T in either one or both of their endpoints. Let E denote the set of half-edges of associated 1-cells of 2-cells in A having an endpoint in T . Note that the other endpoint of such a half-edge belongs to f −1 (x) which is disjoint from Z . We orient each half-edge in E from the endpoint in f −1 (x) to the endpoint in T , so that f respects orientation on the half-edges in E.
Suppose that e ∈ E. Then e is a half-edge of an associated edge of a 2-cell α ∈ A . Thus (with a suitable choice of orientation) the distinguished attaching path R for α has an initial segment of the form e .P.(e ) −1 , where P is an edge-path in T and e is a half-edge of an edge e in f −1 (e) which is not contained in T . Now e is the associated 1-cell of a 2-cell in A , since otherwise it is contained in Z and hence in T . Since e has an end-point in T we have e ∈ E. Note that e is uniquely determined by e . We call e the successor of e and write e = σ(e ). Thus σ : E → E is a well-defined map.
Here are some remarks about this map σ.
(1) σ(e ) = e . For suppose that Q := e .P.(e ) −1 is a subpath of the attaching path R of a 2-cell α ∈ f −1 (α). Since R is cyclically reduced, it follows that P is non-empty and Q is not the whole of R . Moreover f * (Q) is an initial segment of R ±n but is not a power of R, and so f * (Q) > 1 in G. However, P is a closed path in T and by hypothesis f This gives a contradiction. (2) Suppose that e ∈ E, and that e and e = σ(e ) are not half-edges of the same edge. Then the initial segment Q = e .P.(e ) −1 of R = ∂α in the definition is proper and nonempty. If f (Q) = R k for some k then 0 < k < n and so e is a half-edge of another low edge of α . Since it is also the associated 1-cell of a 2-cell α ∈ f −1 (α), and since it cannot be a low edge of two distinct 2-cells, we must have α = α and k ≡ 0 mod n, contrary to assumption. Hence f (Q) is not a power of R. As in Remark 1 above we have f * (Q) > 1 in G. (3) Since E is finite, any chain of the form e 1 , e 2 = σ(e 1 ), e 3 = σ(e 2 ), . . . in E must contain a loop. Without loss of generality let us suppose that e n = e 1 for some n. If, for each pair e j , e j+1 in this sequence, e j and e j+1 are half-edges of associated 1-cells of distinct cells of f −1 (α), then by Remark 2 we have paths This contradiction shows that, if E is not empty, there must be a 2-cell α ∈ A and a half-edge e of the associated 1-cell of α , such that σ(e ) is also a half-edge of the associated 1-cell of α . (4) By assumption, T = Y and so E is non-empty. By Remark 3 there exist α ∈ A and e ∈ E such that e and e := σ(e ) are half-edges of the associated 1-cell e of α . By Remark 1 we cannot have σ(e ) = e , so e and e are the two distinct halfedges of e. In this case the attaching path R of α is a power of Hence e is an almost-free face of α , and T ∪ e ∪ α is a subcomplex of Y which admits a simple low-edge almost-collapse to T .
This gives us the desired contradiction to the maximality of T in T and completes the proof Proof. As in Theorem 3.1, let Z be formed from Y by deleting all the 2-cells of f −1 (α) together with their associated 1-cells. Suppose first that some component T of Z has zero first Betti number. Then there is a sequence of low-edge almost-collapses transforming Y to T . Necessarily this sequence of almost-collapses involves every 2-cell in f −1 (α). Suppose that f −1 (α) = {α 1 , α 2 , . . . , α N } and that α j is attached along a q(j)'th power for each j (where q(j) ≥ 1). Suppose also that p|q(j) for j ≤ J. Then π 1 (Y ) is the free product of π 1 (T ) and the cyclic groups Since p|q(j) for j ≤ J, at least J of these direct factors are isomorphic to F , and the result follows in this case.
Hence we may assume that every component of Z has positive first Betti number. Suppose that there are K components in Z , and N 2-cells in f −1 (α), of which J are attached along p'th powers. Let Z := Z ∪ Y (1) . Then Z is obtained from Z by adding N 1-cells, of which K −1 are required to make Z connected, and the remaining N −K +1 contribute to the first Betti number. So Z has first Betti number at We next consider immersions f : Y → Y n where n > 1. The 2-cell α n of Y n has attaching map R n , where R is not a proper power. It follows that each 2-cell α ∈ f −1 (α n ) has an attaching map of the form S p , where S is not a proper power, f (S) = R q , and pq = n. Theorem 3.3. Let n > 1 and suppose that f : Y → Y n is an immersion, where Y is compact and connected, with first Betti number β. Suppose that none of the 2-cells in f −1 (α n ) is attached by an n'th power. If Y has no free edges, then the number of 2-cells in f −1 (α n ) is bounded above by 5β.
, together with all their low edges. Note that there are at least 2 distinct low edges for each 2-cell in f −1 (α n ), since its attaching map is not an n'th power. Now let C be a component of Z with first Betti number 0. (We call such a component a treeoid.) Then (ψ n • f ) * (π 1 (C)) is a finitely generated subgroup of G with first Betti number 0, and hence trivial.
and there are no 2-cells in f −1 (α n ). So we may assume that C = Y . Since Y is connected, there must be one or more low-edges of 2-cells in f −1 (α n ) that meet C.
We claim that C is incident to at least 5 half-edges of low edges of 2-cells of f −1 (α n ).
Arguing as in the proof of Theorem 3.1, for each such half-edge e there is another half-edge σ(e ) and a cyclic subpath of the attaching path of some α ∈ f −1 (α n ) of the form Q := e · P · σ(e ) −1 with P a path in C. Then f (Q) is an initial segment of the attaching path Still arguing as in the proof of Theorem 3.1, any chain of half-edges e 1 , e 2 , . . . of low edges of cells in f −1 (α n ) that are incident at C with σ(e i ) = e i+1 must contain a loop, and the only possibility for such a loop is a pair e 1 , e 2 with e 1 = σ(e 2 ), e 2 = σ(e 1 ), and f (Q) = R ±1 . Now the path Q = e 1 · P · e −1 2 must contain a high edgeê of α . Sincê e is not a free edge of α , there must be another subpath P in C of the attaching path of a 2-cell α ∈ f −1 (α n ) that passes throughê. Note that P is a cyclic subword of R ±1 .
It is not possible for P to begin at the edge e 1 , else there would be a loop formed from an initial segment P 1 of P to the edgeê together with the initial segment P 1 of P from e 1 to the edgeê. Since f * (C) = 1, it follows that f * (P 1 ) = f * (P 1 ) = U = V −1 . By uniqueness of low and high edges, it then follows that e 1 andê are low and high edges respectively of α , and hence that α = α and so P = P , a contradiction. Essentially the same argument shows that P does not begin or end at either of the edges e 1 , e 2 .
It follows that C is incident to at least 4 distinct half-edges of low edges of 2-cells in f −1 (α n ), namely e 1 = σ(e 2 ), e 2 = σ(e 1 ), e 3 and e 4 . Suppose that these are the only 4 half-edges of low edges of 2cells in f −1 (α n ) incident at C. Then σ(e 4 ) ∈ {e 1 , e 2 , e 3 } and σ(e 3 ) ∈ {e 1 , e 2 , e 4 }. There are essentially two cases. Case 1. Suppose first that σ(e 4 ) = e 1 . Then there is a path Q := e 4 · P · e −1 1 with P in C, such that f (Q ) is an initial segment of the attaching path R ±n of the 2-cell α n .
Recall that Q = e 1 · P · e −1 2 and Q := e 3 · P · e −1 4 = Q, where P, P are paths in C that pass through the high edgeê. Let Q 1 denote the Then Similar arguments yield contradictions in the case where σ(e 4 ) = e 2 , and in the cases where σ(e 3 ) ∈ {e 1 , e 2 }.

Case 2.
We are now reduced to the case where σ(e 3 ) = e 4 and σ(e 4 ) = e 3 . As was the case for Q, e 3 and e 4 are low edges of the same 2-cell. and there is a path in C joining them labelled R ±1 . But the path P also joins these half-edges, and so the label on Q is R ±1 and the 2-cell concerned is α . Arguing as in the case ofê, Q contains a high edge e of α . Since e is not a free edge of α , it is contained in the attaching path of a 2-cell of f −1 (α n ) that meets C but not e 3 or e 4 . Thus e 1 or e 2 is a low edge of this other 2-cell, and so this other 2-cell must be α . Henceê is contained in Q , while e is contained in Q. Let L 1 denote the subpath in Q from the midpoint ofê to e 1 , L 2 the subpath of Q from e 1 to the midpoint of e , L 3 the subpath of Q from the midpoint of e to e 3 , and L 4 the subpath of Q from e 3 to the midpoint ofê. Let and h 3 ≤ h −1 2 . Moreover these inequalities are strict becauseê = e since these are high edges of distinct 2-cells. Thus h : In all cases we have obtained a contradiction, and so C meets at least 5 half-edges of low edges of 2-cells in f −1 (α n ), as claimed.
To complete the proof, suppose that Z has M 0 treeoid components and M 1 non-treeoid components. And suppose that there are The J low edges split into 2J half-edges, of which each treeoid component of Z meets at least 5, as we have seen. Hence 5M 0 ≤ 2J. Combining this with inequality (1) and the fact that J ≥ 2K, we have 5β > 5J − 5K − 5M 0 ≥ 3J − 5K ≥ K as required.
Corollary 3.4. Let g : Y → Y n be an immersion, where n > 1. Suppose that Y is compact and connected, that no 2-cell in g −1 (α n ) has a free edge, and that π 1 (Y ) can be generated by k elements. Then the number of 2-cells in g −1 (α n ) is at most 11k.
Proof. Let f := ψ n • g : Y → Y . Then f is a branch map, and satisfies the conditions of Theorem 3.1. Let p be a prime factor of n > 1 and let F be the field of order p. Then the dimension over F of H 1 (Y , F ) ∼ = π 1 (Y ) ab ⊗ Z F is less than or equal to k, since π 1 (Y ) can be generated by k elements. By Theorem 3. Now we are ready to prove our main result, Theorem A. It is easy to see that the general case of the theorem reduces to the two-factor case, where |Λ| = 2. We restate and prove it in that form. Proof. Let H be a finitely generated subgroup of Γ. We want to show that H is finitely presentable. We proceed by induction on the number of generators for H: clearly it is true for subgroups with at most 1 generator. We can assume that H is generated by k elements, and that every subgroup of G generated by strictly fewer than k elements is finitely presentable. In particular, if H decomposes as a free product, then each free factor (and hence also H) is finitely presentable, using Grushko's theorem. So we may assume that H is freely indecomposable.
Following Peter Scott ([16, Lemma 2,2]), there is a finitely presented group K and an epimorphism F : K → H which does not factor through any non-trivial free product, i.e. such that for any factorisa- Construct a 2-complex Y (resp Y ) with fundamental group Γ (resp. G := (A * B)/ S ) as follows. Let X A , X B be presentation 2-complexes for A, B respectively. Add a 1-cell e to X := X A X B joining the basepoints of the two components, and then attach a 2-cell α along σ n (resp. σ), where σ is a path in X (1) ∪ e representing S ∈ A * B = π 1 (X ∪ e). Thus Y is a simple enlargement of X, and Y → Y is the n-fold branched cover of Y , as defined in §2.
Let W 0 be a finite 2-complex with π 1 (W 0 ) = K, let q H : Y H → Y be the covering of Y with π 1 (Y H ) = H, and subdivide the 1-and 2cells so that the map F : K → H is induced by a combinatorial map g 0 : W 0 → Y H . Now g 0 is not in general an immersion, but by Lemma 2.4 it can be factored as Y H is an immersion. We may assume that Y 0 is minimal (fewest cells) with all of these properties (i.e. finite, indecomposable, π 1 -surjective immersion to Y H ).
Note that if an edge in a connected 2-complex Z is not in the boundary of a 2-cell, then either: 1) the edge is separating and π 1 (Z) decomposes as a free product, or one of the components is simply connected.
2) the edge is non-separating and π 1 (Z) is either infinite cyclic or decomposes as a free product with an infinite cyclic Z factor. By our choice of K, π 1 (Y 0 ) does not decompose as a free product. By our minimality assumption, no edge of Y 0 separates with one component of the complement being simply-connected. Hence neither of the above possibilities 1) or 2) can happen for Z = Y 0 . It follows that every 1-cell of Y 0 occurs at least once in the boundary of a 2-cell.
Note in addition that an edge e of Z that occurs exactly once in the boundary of exactly one 2-cell α, gives rise to an elementary collapse Z Z = Z − {Int α, Int e}; thus again the subcomplexZ of Z is such that π 1 (Z) = π 1 (Z), contradicting the minimality of Y 0 .
It follows by the above remarks that, under the minimality assumption on Y 0 , every edge occurs at least twice in the boundaries of the 2-cells (i.e. either at least once in the boundaries of two different 2-cells or at least twice in the boundary of one 2-cell).
If f 0, * : π 1 (Y 0 ) → H is not an isomorphism, then there are nontrivial loops in Y 0 labelled by words representing non-trivial elements in ker f 0, * .
Choose a sequence of such words w 0 , w 1 , . . . which together normally generate ker f 0, * , and a van Kampen diagram φ j : D j → Y H for each w j = 1 over A * B R (we have fixed presentations for A and B). We can suppose that the words and diagrams are chosen so that each D j as well as being reduced and minimal, is a topological disc.
We can also suppose that the set of 2-cells labelled R, together with all 0-cells and 1-cells incident to them, together with the boundary ∂D j is connected -call this the R-subcomplex of D j .
This last property is one of the essential properties established in the proof of the Freiheitssatz for one relator products of locally indicable groups in [3,9,17], that the natural map A → A * B R is an injection. If the above R-subcomplex is not connected, then there is a subdiagram of D 0 containing some regions labelled R, and whose boundary is a word in A or in B. This subdiagram can be replaced by an A -diagram (or a B-diagram) without regions labelled R. The Freiheitssatz (Theorem 2.1) implies that for each i the word w i labelling the boundary δ i contains non-trivial words in A and in B.
As the diagram is a disk every edge in the interior of D i lies in the boundary of at least two 2-cells.
Form the adjunction space W 1 = Y 0 ∪ ∂D 0 by identifying δ 0 = ∂D 0 with its image in Y 0 , so that the maps f 0 , φ 0 combine to give a map W 1 → Y H . Applying Lemma 2.4 once more, this factors as W 1 → Y 1 Y H where W 1 → Y 1 is surjective and π 1 -surjective, and Y 1 Y H is an immersion. The map Y 0 → W 1 → Y 1 is also an immersion, since the immersion Y 0 Y H factors through it. Each edge in Y 0 occurs at least twice in the attaching maps of 2cells, and hence the same is true of its image in Y 1 by the immersion property. Each interior edge of D 0 has at least two occurrences in the boundaries of 2-cells of D 0 . These map to distinct occurrences of its image in Y H in boundaries of 2-cells, since the diagram D 0 → Y H is reduced. Since each edge of Y 1 is the image of an edge of Y 0 or of an interior edge of D 0 , it follows that no 2-cell of Y 1 has a free edge.
Finally note that w 0 ∈ Ker(π 1 (Y 0 ) → π 1 (Y 1 )). Continuing in this way, if we have inductively defined Y i , and immersions Y 0 · · · Y i , such that no Y j has a free edge and w 0 , . . . , w i−1 ∈ Ker(π 1 (Y 0 ) → π 1 (Y i )) then we can adjoin the diagram D i to Y i and factor the resulting map to Y H through an immersion, giving a new 2- We obtain in this way a sequence of immersions is π 1 -surjective and w 0 , . . . , w i ∈ Ker(π 1 (Y 0 ) → π 1 (Y i+1 )). Hence the sequence of immersions (2) induces a sequence of epimorphisms of fundamental groups that converges to H, and by construction, no 1-cell of Y i appears less than twice in the boundaries of 2-cells. Now let Z i ⊂ Y i be the R-subcomplex made up of the union of the image of Y 0 in Y i together with all preimages of the R-disk α and all of their incident 0-and 1-cells. In other words, Z i is the union of the images of Y 0 and of the R-complexes of the van Kampen diagrams , the number of preimages of α in Y i is bounded above by 11k, where k is the minimal number of generators of H. Since every 1-cell in Y i is incident at a 2-cell, the number of preimages of e in Y i is also bounded (for example, by 11k where is the length of R as a path in Y (1) ).
But every cell in Z i is either in the image of the compact 2-complex Y 0 , or is incident to one of the boundedly many 2-cells that are preimages of α. Hence the Z i are bounded in size, so some subsequence {Z σ(i) } of the Z i stabilises, in the sense that the sequence Z σ(1) Z σ (2) ... consists of isomorphic 2-complexes. Now Wise [18] has shown that any immersion from a finite complex to itself must be an isomorphism. (Otherwise some power of the immersion is a retraction onto a proper subcomplex, contrary to the definition of immersion; we are grateful to Lars Louder for explaining this to us.). Hence in fact we may assume that Z σ(1) Z σ (2) ... consists of isomorphisms. Using this, we identify each Z σ(i) with Z σ(1) via the inverse isomorphism. Now any element of the kernel of the map π 1 (Z σ(1) ) → π 1 (Y H ) = H becomes trivial in some π 1 (Y σ(i) ). But Z σ(1) = Z σ(i) is a subcomplex of Y σ(i) , so any such element can be expressed as the boundary label of a van Kampen diagram ∆ in Y σ(i) . As mentioned earlier in the proof, the R-subcomplex of ∆ may be assumed to be connected, by Theorem 2.1. Thus its complement consists of a finite number of open disks in ∆ whose boundaries are paths in Z is compact, and each of A, B is coherent, it follows that Ker(π 1 (Z σ(1) ) → H) is normally generated by a finite number of loops in Z For sufficiently large N , these loops are all nullhomotopic in Y σ(N ) , and it follows that π 1 (Y σ(N ) ) → H is an isomorphism. Since Y σ(N ) is compact, H is finitely presentable, as claimed.
We end this section by noting that stronger versions of Theorems E and F hold in the torsion case. Translated into our set-up, a result in [7] says the following.
Theorem 3.6 ([7], Theorem 3.3). Let Y := X ∪ e ∪ α be a simple enlargement where X is a 2-complex, all of whose fundamental groups are locally indicable, e is a 1-cell, and α is a 2-cell. Let n > 1 be an integer, and let Y n → Y be the n-fold cyclic branched cover defined at the start of this section. Let P : Σ → Y n be a reduced picture on a compact orientable surface Σ that contains V α-discs. Then the number of points where P meets ∂Σ is at least where G λ are locally indicable groups, R ∈ * λ G λ is cyclically reduced of length at least 2, and n > 1 is an integer. Then (1) If λ, µ ∈ Λ and g ∈ G are such that the intersection G λ ∩ G g µ in G is non-trivial, then µ = λ and g ∈ G λ .
(2) Any free abelian subgroup of G of rank greater than 1 is contained in a conjugate of G λ for some λ ∈ Λ. (3) Any subgroup of G that is isomorphic to a RAAG based on a connected graph with at least one edge is contained in a conjugate of G λ for some λ ∈ Λ.
Proof. Without loss of generality we can assume that R is not a proper power in * λ G λ . We can reduce to the case of a two-factor one-relator product (A * B)/ R n by choosing a 2-part partition Λ = Λ(1) Λ(2) with the property that each Λ(j) contains a λ(j) such that R contains a letter from G λ(j) , then setting A := * λ∈Λ(1) G λ , B := * λ∈Λ(2) G λ . We then form a simple enlargement Y = X ∪ e ∪ α of X := X A X B where X A , X B are connected 2-complexes; π 1 (X A ) ∼ = A and π 1 (X B ) ∼ = B; e is a 1-cell joining the base points of X A and X B , and α is a 2-cell attached along a path representing R ∈ A * B. The result follows by applying Theorem 3.6 to pictures over Y n as follows.
(1) A conjugacy relation y = x g with x ∈ G λ and y ∈ G µ can be expressed using a reduced picture P : Σ → Y n where Σ is an annulus and the components of ∂Σ map to reduced paths in X which represent x, y respectively. In particular these paths do not involve e, and so no arc of P meets ∂Σ. Since χ(Σ) = 0, it follows from Theorem 3.6 that P has no α-discs. So we can regard P as a picture over X ∪e, and the conjugacy relation y = x g already holds in the free product π 1 (X ∪ e) = A * B = * λ G λ . The result follows from well-known properties of conjugacy in free products. (2) A commutator relation xy = yx in G can be expressed using a reduced picture P : Σ → Y where Σ is the torus. Since χ(Σ) = 0 and ∂Σ = ∅, Theorem 3.6 again tells us that P has no α-discs.
Thus P is a picture over X ∪ e, and the commutator relation xy = yx already holds in π 1 (X ∪ e) = A * B = * λ G λ . The result again follows from well-known properties of commutation in free products. (3) The final assertion follows easily from the first two. The subgroup corresponding to any edge is contained in some G g λ , and the subgroup corresponding to any vertex can be contained in at most one G g λ , so there is a unique λ ∈ Λ and a unique right coset G λ g such that G g λ contains the RAAG in question.

The Torsion-free Case and the Betti number property
In this section we consider one-relator products of locally indicable groups in which the relator is not a proper power. We first note that Theorem 3.1 is stronger in this case.
Theorem 4.1. Let X be a 2-complex such that every connected component of X has locally indicable fundamental group, and let Y be a simple enlargement of X. Suppose that f : Y → Y is an immersion, with Y compact and connected, and let A := Y f −1 (X). Let Z be the subcomplex obtained from Y by removing all the 2-cells in A and their associated 1-cells. Suppose also that, for some component T of Then Y collapses to T through a sequence of low-edge collapses.
Proof. The map f is an immersion, hence a branch-map with no branch-points. By Theorem 3.1, Y can be transformed to T through a sequence of low-edge almost-collapses. But since there are no branch points of f in Y , each low-edge almost-collapse is a genuine collapse. The result follows.
Proof of Theorem B. Case 1: Y = X ∪ e where e is a 1-cell. Let f : Y → Y be an immersion with Y compact and connected, and suppose that χ(Y ) > 0. Let X := f −1 (X). Since X has the non-positive immersions property, each component of X either is contractible or has non-positive Euler characteristic. In particular each component of X has Euler characteristic at most 1. Since Y is connected and constructed from X by attaching 1-cells, it follows that Y has Euler characteristic at most 1, and hence by hypothesis χ(Y ) = 1. But χ(Y ) = 1 implies that each component of X has Euler characteristic 1 (and hence is contractible) and that these components are connected in a tree-like manner by the edges in f −1 (e) to form Y . Hence Y is also contractible, as claimed. e is a 1-cell and α is a 2-cell.
Let f : Y → Y be an immersion with Y compact and connected, and suppose that χ(Y ) > 0. Let Z be as in Theorem 4.1. Since Z is obtained from Y by removing equal numbers of 1-and 2-cells, we have χ(Z ) = χ(Y ) > 0, and hence there is a component T of Z with χ(T ) > 0. But f (T ) ⊂ X ∪e and X ∪e has the non-positive immersions property by Case 1 above, so T is contractible. By Theorem 4.1, Y collapses to T , so Y is also contractible, as claimed.
Proof of Theorem D. Let K be a finitely generated subgroup of π 1 (Y ). Suppose that K has first Betti number m and second Betti number β 2 (K) ≥ n := m + F (m − 1). Suppose that K can be generated by d elements (where of course d ≥ m). We will show that K = 1, from which the assertion of the Theorem follows.
We can construct an epimorphism K → K for some group K which has first Betti number m and possesses a d-generator, (d − m)-relator presentation P. Let F be the free group on these d generators, and R the kernel of the epimorphism F K K. Then we can choose relations r 1 , . . . , r n ∈ R ∩ [F, F ] which are linearly independent in and add them to the presentation P to get a new presentation P of a group K , also admitting an epimorphism to K. This epimorphism can be realised by a combinatorial map f : V → Y for some subdivision V of P . Lemma 2.4 gives a factorization of f as g •f , where g : Y Y is an immersion, andf : V → Y is a surjective combinatorial map which is also surjective on π 1 , and hence also on H 1 . Hence Y is connected with first Betti number β 1 (Y ) ≤ m. On the other hand, since g * (π 1 (Y )) = f * (π 1 (V )) = K has first Betti number m, we also have β 1 (Y ) ≥ m and hence β 1 (Y ) = m. Now the map Z n ∼ = H 2 (V ) → H 2 (K ) → H 2 (K) is injective and factors through H 2 (Y ). Hence Y has second Betti number ≥ n.
For each component X j of X := g −1 (X), embed X j (for example via a mapping cylinder construction) into a classifying space X j for the subgroup K j := g * (π 1 (X j )) of K, in such a way that the embedding map realises the given map g * on fundamental groups. Let X denote the disjoint union of these X j for all the components X j of X . Form Z and Y from Z , Y respectively by adjoining X along X . We can extend the map g : Y → Y to a map from Y to a classifying space Y for π 1 (Y ), in such a way that the restriction to each X j factors through the covering of Y corresponding to K j . By hypothesis, each non-trivial K j has the property that Since X j is a classifying space for K j , it follows that each non-contractible X j satisfies Without loss of generality, suppose that there are J non-contractible components X j , 1 ≤ j ≤ J of X, and C contractible components X c , J + 1 ≤ c ≤ J + C. Then Now Z is constructed from X by adding a finite number -say -of 1-cells. Then Y is constructed from Z by adding equal numbers of 1and 2-cells. It follows that (4) Since Y is constructed from X by adding 1-cells and 2-cells, we have H 3 (Y , X) = 0, and so H 2 (X) → H 2 (Y ) is injective, from the long exact homology sequence. In particular β 2 (Y ) ≥ β 2 (X).
From (4), recalling that Y is connected, we have and as F is non-decreasing we obtain We assumed that F is not a second Betti bound for π 1 (Y ), so from (3) and (4) we derive But F is non-decreasing and It follows that C > . Thus there is a component T of Z such that the number k T of contractible components of X contained in T is strictly greater than the number T of 1-cells of Z \ X contained in T . But T is connected, and it follows that k T = T + 1, and that T is also contractible. Hence T := T ∩ Z is a component of Z such that g * (π 1 (T )) = {1} in π 1 (Y ). By Theorem 4.1, Y collapses to T , so K = g * (π 1 (Y )) = 1 in π 1 (Y ). This completes the proof.
We end this section with a slightly strengthened version of the Freiheitssatz, and an application to complexes which immerse into simple enlargements.
Lemma 4.2. Let Y := X ∪e∪α be a simple enlargement of a 2-complex X, where e is a 1-cell and α is a 2-cell, and where every fundamental group of X is locally indicable. Let φ : ∆ → Y be a reduced van Kampen diagram which is either spherical or a disk diagram. If φ −1 (α) = ∅ then ∆ is a disk diagram, and there is a 2-cell α mapping to α such that the low-edge of α lies on the boundary of ∆.
Proof. Suppose not. Then there is a sequence of 2-cells α 1 , . . . , α n ∈ φ −1 (α) such that for each j the low edge e j of α j is also on the boundary of α j+1 (indices modulo n). By definition of low edge, either of the two paths P j+1 in ∂α j+1 from the midpoint of e j to that of e j+1 represents an element U j+1 = [φ(P j+1 )] < 1 in π 1 (Y ) (with respect to a fixed left ordering of π 1 (Y )). Hence their product But this contradicts the fact that U is represented by φ(P ) where P := P 1 · P 2 · · · P n is a closed path in the simply-connected space ∆. (1) The inclusion-induced map π 1 (Z j ) → π 1 (Y ) is injective for each j ∈ J; and (2) π 2 (Y ) is generated as a Zπ 1 (Y )-module by the images of π 2 (Z j ) for all j ∈ J. Proof.
(1) Apply Lemma 4.2 to a reduced van Kampen diagram in the plane, with boundary labelled by a path in Z j representing an element w ∈ Ker(π 1 (Z j ) → π 1 (Y )). Since the low edges of 2cells in f −1 (α) are by definition excluded from Z , the Lemma says that there are no 2-cells of is injective, and so the boundary label w of ∆ is already trivial in π 1 (Z j ).
(2) Applying Lemma 4.2 to the case where ∆ is a spherical diagram, we see that no 2-cell of ∆ maps to α under f • φ. It follows that π 2 (Y ) is generated as a Zπ 1 (Y )-module by the image of π 2 (Y \ f −1 (α)). Since Y \ f −1 (α) is formed by adding 1-cells to Z , we deduce that in fact π 2 (Y ) is generated as a Zπ 1 (Y )module by the images of π 2 (Z j ) for all j ∈ J.

Z 3 , wreath products and Thompson's group as subgroups
In this section we prove the first three parts of Theorem E.
Proof of Theorem E. Suppose without loss of generality that R = S m for some m ≥ 1, where S is not a proper power.
We use Theorem 2.2 and Shapiro's Lemma as in the proof of Corollary 2.3 (using Z coefficients) giving Since K has trivial intersection with any conjugate of the finite subgroup S , it follows that there is precisely one conjugate G g λ of precisely one free factor group G λ such that H r (K; Z) = H r (K ∩ G g λ ; Z). In particular K ∩ G g λ has index 1 in K, that is K < G g λ .
The commutator subgroup [K, K] of K is free abelian of infinite rank, with basis {x n , n ∈ Z}, and K = [K, K] t where tx n = x n+1 t for all n. By Case 1 with r = 3, there is a unique G g λ such that x 1 , x 2 , x 3 ∈ G g λ . For any other n ∈ Z, we can apply Case 1 with r = 4 to get a unique G h µ containing x 1 , x 2 , x 3 , x n . But then uniqueness of g, λ gives us h = g and µ = λ. Hence [K, K] < G g λ . But then the normality of There is a presentation of K of the form Let y n := x −1 n x n+1 . Then y m commutes with x n and y n in F if m + 1 < n. Hence x 2 1 and y 2 generate a copy of Z wr Z in F , so by Case 2 there is a unique λ and a unique coset G λ g such that x 2 1 and {y 2k ; k ≥ 1} all belong to G g λ . For odd k, the elements y k , y k+3 , y k+5 , y k+7 generate a free abelian subgroup of F of rank 4, so they belong to a unique G h µ by Case 1 with r = 4. But applying Case 1 with r = 3 to y k+3 , y k+5 , y k+7 gives us g = h and λ = µ. Hence y n ∈ G g λ for all n. Finally, we can apply the same argument to the free abelian subgroup with basis {y 1 , y 3 , y 5 , x 7 } to deduce that x 7 ∈ G g λ . Since F is generated by x 7 together with the y n , it follows that K < G g λ as claimed.

A new proof of Brodskiȋ's Lemma
Proof of Theorem F. By Theorem 3.7 we may assume that G is torsionfree, that is that the relator R is not a proper power in * λ G λ . By Corollary 2.3 we know that the intersection G λ ∩ G g µ is free unless the conclusion of the theorem holds. So let us suppose that there is a free subgroup of rank 2 in G λ ∩ G g µ . As in the proof of Corollary 3.7, we reduce to the case where |Λ| = 2, so that G is a one-relator product of two locally indicable groups A, B. Then we model the situation geometrically as follows.
Let Y = X∪e∪α be a simple enlargement of X := X A X B where X A and X B are connected 2-complexes with locally indicable fundamental groups A, B respectively. As usual e is a 1-cell and α is a 2-cell attached along a path representing R ∈ π 1 (X ∪ e) ∼ = A * B ∼ = * λ G λ .
Suppose that K is a free subgroup of A of rank 2 and P is either a loop at the base point of X A or a path from the base point of X A to the base-point of X B , such that P −1 KP is equal in π 1 (Y ) to a subgroup of A or of B respectively. Let V = S 1 ∨ S 1 be a 2-petal rose with base-point * , and let φ : V × [0, 1] → Y be a map representing this conjugacy set-up. That is, φ(V × {0}) is a pair of curves in X A that generate K, φ(V × {1}) is a pair of curves in X, and φ({ * } × [0, 1]) is the path P .
After subdividing cells in V × [0, 1] to make φ be cellular, we can apply Lemma 2.4 to factor φ as a surjective, π 1 -surjective map φ : V × [0, 1] → Y followed by an immersion f : Y Y . Let α denote the 2-cell in Y \ X, Z the subcomplex of Y obtained by deleting every 2-cell in f −1 (α) along with its low edge, and X := f −1 (X) ⊂ Z .

Now as well as being
is π 1 -surjective and factors through X 0 , the inclusion X 0 → Y is also π 1 -surjective. Thus X 0 has first Betti number ≥ 2. The same applies to the component X 1 of X that contains φ (V × {1}).
No component of Z has first Betti number 0. For otherwise Y collapses onto that component, by Theorem 4.1, and has first Betti number 0, a contradiction. Since Y is connected with first Betti number 2, and is formed from Z by attaching equal numbers of 1-and 2-cells, it follows that all but at most one of the components of Z have first Betti number 1, and no component has first Betti number greater than 2.
Since Z is formed from X by attaching 1-cells, it follows in turn that at most one component of X can have first Betti number greater than 1. Hence X 1 = X 0 .
is a path in Y connecting two points of X 0 , and P = f (P ), so both ends of P lie in X A . Thus P is a loop in Y based at the base-point of X A . Moreover, since X 0 is connected and X 0 → Y is π 1 -surjective, there is a path P 0 in X 0 with the same endpoints as P , and a loop P in X 0 that is homotopic rel base point in Y to P · P −1 0 . Thus the element of G represented by P is g := [P ] = [f (P )] = [f (P · P 0 )] ∈ π 1 (X A ) = A.

Direct Products of Free Groups as subgroups
In this section we prove part four of Theorem E.

Proof.
As in the proof of Theorem F, we may assume that |Λ| = 2, and model the situation using a simple enlargement Y = X ∪ e ∪ α of X := X 1 X 2 where X 1 and X 2 are connected 2-complexes with locally indicable fundamental groups G 1 , G 2 respectably. As usual e is a 1-cell and α is a 2-cell attached along a path representing R ∈ π 1 (X ∪ e) ∼ = G 1 * G 2 = * λ G λ , and G = π 1 (Y ).
Suppose that a, b, c, d ∈ G generate K, a direct product of two free groups a, b and c, d of rank 2.
We shall show that K is contained a conjugate of one of the factors G λ in G.
Let V = S 1 ∨ S 1 be a 2-petal rose with base-point * . There is a map φ : V × V → Y , combinatorial after subdivision, with φ * (π 1 (V × V )) = K < G. The restriction to any one of the four subcomplexes of V × V isomorphic to S 1 × S 1 is a toral picture over Y representing one of the four commutator equations ac = ca etc. We may assume without loss of generality that each of the four pictures is reduced.
Hence we may assume that R is not a proper power. Using Lemma 2.4, we factor φ through an immersion f : Y → Y such that V × V → Y is surjective and π 1 -surjective. In particular π 1 (Y ) ∼ = F 2 × F 2 and f : Y → Y is π 1 -injective, since φ is π 1 -injective. Thus Y is a connected 2-complex, with first Betti number equal to 4 and second Betti number at least 4, so χ(Y ) > 0. Let Z ⊂ Y be the subcomplex in Theorem 4.1.
Then Z is obtained from Y by the removal of equal numbers of 1-and 2-cells, so χ(Z ) = χ(Y ) > 0, and some component T of Z has positive Euler characteristic. By Theorem 4.1 we may assume that f * (π 1 (T )) = {1} in G, for otherwise Y collapses to T implying that f * (π 1 (Y )) = f * (π 1 (T )) = {1} in G, contrary to hypothesis. Now f (T ) ⊂ X ∪ e. Since π 1 (X ∪ e) = * G λ , the image K of π 1 (T ) in π 1 (X ∪ e) splits as a free product * K j , where each K j is either cyclic or contained in a conjugate of one of the G λ . Suppose that some K j is contained in G g λ . But as the factor groups are locally indicable, G λ embeds in G via the natural map, and hence K j is isomorphic to a finitely generated subgroup of F 2 × F 2 . Moreover, it follows from Lemma 4.2 that f | T is π 1 -injective, and π 1 (T ) ∼ = K j is isomorphic to a finitely presented subgroup of F × F contained in some G λ . Observe that a finitely generated subgroup L of F 2 × F 2 either contains a copy of F 2 × F 2 or has Euler characteristic ≤ 0 (and is Z, Z × Z or free or Z × f ree). This follows easily by considering the restrictions to L of the coordinate projections p 1 , p 2 : F 2 × F 2 → F 2 . If either is injective, then L is free. If p 1 has rank 1 kernel, then there is an exact Finally, if ker(p 1 ) and ker(p 2 ) are both (free) of rank ≥ 2, then The components of Z with fundamental groups that are free products of free groups and Z × f ree groups have non-positive Euler Characteristic. But we have seen that there is at least one component with positive Euler characteristic, so some π 1 (T ), and so some G g λ , contains a subgroup of K isomorphic to F × F . Now any subgroup of K = a, b × c, d that is isomorphic to F 2 ×F 2 has the form L = L 1 × L 2 where L 1 , L 2 are subgroups of a, b , c, d respectively. If L < G g λ then we have F 2 ∼ = L 2 < G g λ ∩ G ga λ and so a ∈ G g λ by Brodskiȋ's Lemma, Theorem F. Similarly b, c, d ∈ G g λ and so K < G g λ .

Right Angled Artin Groups -RAAGs -as subgroups
In this section we prove the fifth part of Theorem E.
Let Γ := Γ(G) be the RAAG defined by a compact connected graph G with more than one vertex and with no cut points. We must show that any subgroup of G isomorphic to Γ is contained in a conjugate of some G λ . In the case where R is a proper power, this follows from Corollary 3.7, so we may assume that R is not a proper power.
We first reduce to the case where G = C n is a cycle of length n. Suppose that the result is true in the case of cycles. Let C denote the set of cycles in G, and say that C 1 ∼ C 2 for C 1 , C 2 ∈ C if C 1 , C 2 have an edge in common. Then by assumption if C 1 ∼ C 2 there are λ 1 , λ 2 ∈ Λ and g 1 , g 2 ∈ G such that Γ(C 1 ) < G g 1 λ 1 and Γ(C 2 ) < G g 2 λ 2 . But any edge in C 1 ∩ C 2 gives rise to a free abelian subgroup of rank 2 in G g 1 λ 1 ∩ G g 2 λ 2 . By Theorem F it follows that λ 1 = λ 2 and g 2 ∈ G λ 1 g 1 . Now let ≈ be the equivalence relation on C generated by ∼, and let G 1 be the union of the cycles in some ≈-class C 1 ⊂ C. Then G 1 is a connected subgraph of G, and it follows from the above discussion that there is a unique λ ∈ Λ and a unique right coset G λ g such that Γ(G 1 ) ⊂ G g λ . Now by definition of G 1 , any cycle containing an edge of G 1 is contained entirely in G 1 . It follows that any path in G containing an edge of G 1 and an edge of G \ G 1 must pass through a cut-point of G. Since G has no cut-points, there is no such path, and it follows that G 1 = G. Hence, as claimed, the general result follows from the special case where G is a cycle.
Suppose then that G = C n is a cycle of length n. By parts 1 and 4 of the theorem we may assume that n ≥ 5. Now Γ has a standard presentation Arguing as in the proofs of Theorem F and part 4 of Theorem E, we reduce to the two-factor case G = (A * B) R and represent this geometrically using a simple enlargement Y = X ∪ e ∪ α where X = X A X B and where A = π 1 (X A ), B = π 1 (X B ), and G = π 1 (Y ) are all locally indicable.
We shall show that any subgroup of G isomorphic to Γ is contained in a unique conjugate of A or B.
Let P be the 2-complex model of the presentation (5), and let φ : P → Y be a map such that φ * : Γ = π 1 (P) → π 1 (Y ) is injective. After first subdividing P to make φ combinatorial, we can apply Lemma 2.4 to factor φ as f • φ , where φ : P → Y is surjective and π 1 -surjective, and f : Y Y is an immersion. In particular Y is compact, and φ is a π 1 -isomorphism since it is π 1 -surjective and φ = f • φ is π 1 -injective. Thus f * maps π 1 (Y ) isomorphically onto a subgroup isomorphic to Γ. Let Z ⊂ Y be the subcomplex in Theorem 4.1 and X := f −1 (X) ⊂ Z . By Theorem 4.1 we may assume that β 1 (T ) = 0 for each component T of Z . By Corollary 4.3, together with the fact that Z is formed by adding 1-cells to X , it follows that each component of X maps π 1 -injectively to Y , and that π 2 (Y ) is generated as a ZΓ-module by the images of π 2 (X , x) for all 0-cells x of Y . We may add cells in dimensions ≥ 3 to each component of X to get a classifying space for its fundamental group. Let X, Z, Y be the union of X (resp. Z , Y ) with all these additional high-dimensional cells. Then Y is a K(Γ, 1)- Since Y is formed from Z by adding equal numbers of 1-and 2-cells, and since each component of Z has positive first Betti number, there exists a component Z of Z such that β 2 ( Z) ≥ β 1 ( Z). Since Z is formed from X by adding only 1-cells, there is a component X of X that is contained in Z, such that β 2 ( X) ≥ β 1 ( X).
Let K := π 1 ( X). Then K is isomorphic to a subgroup of Γ by Corollary 4.3, and H 2 (K) ∼ = H 2 ( X) embeds into H 2 (Γ) ∼ = H 2 (Y ) since X is a component of X and H 3 (Y , X) = 0 (all the higher dimensional cells in Y lie in X). Moreover f * (K) is conjugate in G to a subgroup of A or of B. Without loss of generality we assume that f * (K) < A.
In the standard presentation (5), each generator x j determines a 1cycle in H 1 (Γ), which we will also denote x j . Each relator [x j , x j+1 ] determines a toral 2-cycle τ j ∈ H 2 (Γ). Let S j < H 1 (Γ) and T j < H 2 (Γ) denote the cyclic subgroups generated by x j and τ j respectively.
Then H 2 (Γ) is the direct product of the cyclic groups T j (j = 1, . . . , n) and H 1 (Γ) is the direct product of the cyclic subgroups S j . We consider the images of H i (K) in H i (Γ) for i = 1, 2 and the coordinate projections of these images to the S j and T j . Claim 1. If H 2 (K) projects non-trivially to T j , then there is an element u j of K ∩ x j , x j+2 in which x j has non-zero exponent sum. Similarly there is an element v j+1 of K ∩ x j−1 , x j+1 in which x j+1 has non-zero exponent sum. This means that the image of H 1 (K) in S 1 × · · · × S n contains a rank 2 subgroup of {0} × · · · × {0} × S j−1 × S j × S j+1 × S j+2 × {0} × . . . {0} which projects to the rank 2 subgroup of S j × S j+1 generated by the images of u j and v j+1 . Claim 2. If H 2 (K) projects to a rank 2 subgroup of T j × T j+1 , then the elements u j , v j+2 ∈ K ∩ x j , x j+2 in Claim 1 can be chosen to be linearly independent modulo the commutator subgroup of x j , x j+2 . Consequently K ∩ x j , x j+2 has rank at least 2, whence f * (x j+1 ) ∈ A by Theorem F together with the fact that x j+1 commutes with x j and x j+2 .
We postpone the proofs of the claims, and first show that the result follows from them. By hypothesis b := β 2 (K) ≥ β 1 (K) ≥ 1. We can choose b integers j(1) < · · · < j(b) in the range 1, . . . , n such that H 2 (K) projects to a rank b subgroup of T j(1) × · · · × T j(b) .
If b = n then by Claim 2 we have f * (x j ) ∈ A for all j, so f * (Γ) < A. Now we assume that b < n and derive a contradiction. First suppose that the sequence J := [j(1), . . . , j(b)] contains 2 consecutive numbers (modulo n), then up to a cyclic permutation we may assume that j(1) = 1, j(2) = 2, and j(b) < n. Then by Claim 2, x 2 ∈ K, and the image of H 1 (K) in H 1 (Γ) contains the rank 3 subgroup of S 1 × S 2 × S 3 × S 4 × {0} × · · · × {0} generated by the images of {u 1 , u 2 , v 3 }. If 3 ∈ J then another application of Claim 2 shows that the image of v 4 can be added to this list to give a rank 4 subgroup.
Next assume that J omits two consecutive numbers modulo n. Without loss of generality, suppose that j(1) = 2 and j(b) < n. It follows by induction on b from Claim 1 that H 1 (K) projects to the rank b + 1 subgroup of S 2 × S 3 × · · · × S j(b)+1 generated by the images of u 2 and {v 3 , v j(2)+1 , . . . , v j(b)+1 }, again contradicting the inequality β 1 (K) < b.
Hence we are reduced to the case where n is even, b = n/2 and without loss of generality J consists of the odd numbers 1, 3, . . . , n − 1. But now Claim 1 implies that {u 1 , . . . , u n−1 } generate a subgroup of Π j odd S j of rank at least b − 1, while {v 2 , . . . , v n } generate a subgroup of Π j even S j of rank at least b−1. So β 1 (K) ≥ n−2. But by hypothesis β 1 (K) ≤ b = n/2, so n ≤ 4, which is again contrary to hypothesis.
This completes the proof, modulo Claims 1 and 2.
To prove the claims, note that any element of H 2 (K) can be represented by a 2-cycle in X, hence by a map Ψ : Σ → X for some closed orientable surface Σ. Let Q be a subset of P consisting of: • a single interior point q j of each 1-cell x j ; and • within each 2-cell [x j , x j+1 ], a pair of arcs that cross each other exactly once, one joining the two occurrences of q j in the boundary, and the other joining the two occurrences of q j+1 .
The arcs of Q in [x j−1 , x j ] and in [x j , x j+1 ] that join the occurrences of q j represent the cocycle ξ j ∈ H 1 (Γ) dual to x j : a closed path P in P that is transverse to Q is mapped by ξ j to the algebraic number of times it crosses these two arcs. The integer arising in this way can also be regarded as the projection of the 1-cycle [P ] to the direct factor S j of H 1 (Γ).
In the same way, the intersection point of the two arcs of Q in [x j , x j+1 ] represents the 2-cocycle in H 2 (Γ) dual to τ j : a 2-cycle represented by a map π from an oriented surface to P that is transverse to Q is mapped by this 2-cocycle to the algebraic number of preimages of the intersection point. The integer so arising can be interpreted as the projection of the 2-cycle [π] to the direct factor T j of H 2 (Γ). Now each of P and Y is a K(Γ, 1)-space. So they are homotopy equivalent. Moreover, the fact that Ψ : P → Y is a π 1 -isomorphism means that it is a homotopy equivalence. Choose a homotopy inverse Ψ : Y → P. We may assume that Φ and Ψ are chosen in such a way that the composite Ψ • Φ : Σ → P is transverse to Q. Thus, for each j, the preimage of the arcs of Q that represent the 1-cocycle ξ j is a closed 1-submanifold γ j of Σ. Hence γ j consists of a finite collection of simple closed curves.
The collection of curves γ j may intersect γ j±1 transversely in double points, but does not intersect any of the other γ k . The value of the projection of the 2-cycle [Φ] ∈ H 2 (K) to T j is read off from Σ by counting the algebraic number of crossings of γ j with γ j+1 . Now each closed curve C in γ j may have a sequence of crossings with γ j−1 and/or γ j+1 and these crossings may be positive or negative. A parallel copy C of C will have precisely the same sequence of crossings as C but will be disjoint from γ j . Moreover, C may be deemed to start and end at a point disjoint from j γ j = (Φ • Ψ) −1 (Q).
Since P \ Q is contractible and contains the base-point of P, we may regard (Φ • Ψ)(C ) as an element w of Γ. Indeed w ∈ K since Φ • Ψ factors through X. The element w is represented by the word obtained by reading the sequence of signed crossings of C with γ j−1 and γ j+1 . Clearly this word involves only the generators x j±1 , and so we have w ∈ K ∩ x j−1 , x j+1 .
If H 2 (K) projects non-trivially to T j , then we can choose Φ to represent an element which projects non-trivially. This means that the algebraic number of crossings of γ j with γ j+1 is non-zero. In particular there will be curves C in γ j and D in γ j+1 such that the algebraic intersection numbers of C with γ j+1 and D with γ j are non-zero. The element w obtained by applying the above process to C (resp. D) belongs to K ∩ x j−1 , x j+1 and x j+1 appears with non-zero exponent sum (resp. belongs to K ∩ x j , x j+2 and x j appears with non-zero exponent sum). These words provide the elements v j+1 and u j respectively.
To prove Claim 2, apply the above reasoning to each of a pair of 2-cycles which projects to a linearly independent pair of elements of T j × T j+1 . We can represent these by maps Φ 1 : Σ 1 → X, Φ 2 : Σ 2 → X for different surfaces Σ 1 , Σ 2 . Each curve in the resulting submanifolds γ j+1 in Σ 1 and Σ 2 gives rise to an element of K ∩ x j , x j+2 . Modulo the commutator subgroup of x j , x j+2 this becomes an element (m, n) ∈ Z 2 of the projection of H 1 (K) to S j × S j+2 . Summing these (m, n) over all the γ j+1 curves in Σ 1 (resp. Σ 2 ) gives the projection of [Φ 1 ] (resp. [Φ 2 ]) to T j ×T j+1 . Since these two projections are linearly independent, the collection of vectors (m, n) spans a rank 2 subgroup of Z 2 , and so we choose two words in K ∩ x j , x j+2 arising from the γ j+1 -curves whose images in S j × S j+2 are linearly independent. This completes the proof of the claim, and hence of the theorem.