THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS

INNA BUMAGINA

Abstract. A group is said to be Hopf (or Hopfian) if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopf. Our proof generalizes a theorem of Z. Sela [36].

1. Introduction

A group G is called Hopf if it can not be isomorphic to its proper factor. Finite groups and free groups are examples of groups which are Hopf (the first assertion is obvious, the second one follows from a theorem of Nielsen [22, Chapter 1], alternative proofs can be found in [7]). More generally, by a theorem of Mal’cev, every finitely generated, residually finite group is Hopf [23]. Since every finitely generated subgroup of a residually finite group is residually finite, Mal’cev’s theorem implies that if H is a finitely generated subgroup of a residually finite group G then H is Hopf. Another class of examples of this sort was given by Geoghegan, Mihalik, Sapir and Wise in [15].

On the other hand, there are examples of finitely generated and finitely presented infinite groups that do not have the Hopf property. Such examples were given by B. Neumann [25], G. Higman [16], Baumslag and Solitar [2] (see also [19, Appendix, Part III, §15 ] for the relevant discussion). Meier [24] generalized the examples of Higman and Baumslag and Solitar. Wise [39] showed that even fundamental groups of non-positively curved complexes may be non-Hopf. Sela proved the following remarkable result ([36]).

Theorem 1.1 (Sela). A torsion-free hyperbolic group is Hopf.

The definition of a (word) hyperbolic group is given in Section 2 (Definition 2.6). Hyperbolic groups have many nice properties, but very few of them (like solvability of the word problem in linear time)

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are inherited by finitely generated subgroups. Let us first note that a finitely generated subgroup of a hyperbolic group is not necessarily a hyperbolic group itself: Rips [30] gives an example of a hyperbolic group G that has a subgroup H which is not finitely presented, hence, H is not hyperbolic (see Section 2 for more details). One more important property of hyperbolic groups, which is a consequence of a theorem of Paulin and Rips’ theory, is as follows: if G is a torsion-free hyperbolic group, and the outer automorphism group Out(G) is infinite then G splits into an amalgamated product over a cyclic subgroup. This property is not inherited by subgroups of torsion-free hyperbolic groups; in Section 2.5, we give an example of a subgroup H of a torsion-free hyperbolic group G such that Out(H) is infinite but H has no cyclic splittings. We should also mention that non-elementary, torsion-free, freely indecomposable hyperbolic groups possess the co-Hopf property (which was proved by Sela in [37, Theorem 4.4]). Recall that a group is said to be co-Hopf if every injective endomorphism of the group is surjective. This property is not inherited by non-elementary, freely indecomposable subgroups of torsion-free hyperbolic groups as was shown by Kapovich and Wise in [21].

The question of finitely generated subgroups of hyperbolic groups being Hopf was proposed by Kapovich and Wise [21]. This question is motivated by the question of hyperbolic groups being residually finite. In view of Mal’cev’s theorem, a negative answer would mean that hyperbolic groups are not residually finite in general.

In the present paper, we prove the following generalization of Theorem 1.1.

Theorem 3.5. Finitely generated subgroups of a torsion-free hyperbolic group are Hopf.

Theorem 1.1 does not imply Theorem 3.5. However, we show that a modification of the proof given by Sela in [36], may be applied for subgroups of hyperbolic groups. Our approach is as follows.

Let H = ⟨α_{1}, …, α_{t}⟩ be a subgroup of a torsion-free δ−hyperbolic group Γ, let X = X(Γ, S) be the Cayley graph of Γ with respect to a set S of generators, and let ψ: H ⟶ H be an epimorphism. Our aim is to show that ψ is an isomorphism of H, in other words, that ker(ψ) is trivial. We assume that the kernel of ψ is not trivial which allows us

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 3

to apply the Bestvina-Paulin method (see [3], [27] [28] and Section 2.3 below) and construct a real tree Y endowed with an isometric action of H. The real tree is the limit of a sequence of hyperbolic metric spaces; these metric spaces are obtained from X by rescaling its metric. If ker(ψ) is non-trivial then there is a proper factor group H_{∞} of H that acts faithfully on the real tree, and a canonical quotient map η: H ⟶ H_{∞}. At this point we apply Rips’ theory ([32], [5]) and the canonical JSJ decomposition ([34]), both adapted by Sela to finitely generated groups ([35] and [36]), in order to analyze the algebraic properties of H_{∞}. This analysis allows one to show that the proper quotient map η: H ⟶ H_{∞} is an isomorphism, which is a contradiction.

The paper is organized as follows. In Section 2 we give basic definitions and describe in brief the main ingredients of the proof. In Section 3 we obtain a free decomposition H = H̃_{1} ∗ ⋯ ∗ H̃_{l} ∗ F̃ such that each factor in this free decomposition is mapped by a power of ψ onto a conjugate of itself. We show that ψ is an isomorphism of the whole group H provided ψ|H̃_{i} is an isomorphism for all i = 1, 2, …, l. In Section 4 we adapt the arguments of Sela given in [36, Section 3] in order to prove that ψ|H̃_{i} is an isomorphism.

I am deeply grateful to Professor Zlil Sela for suggesting me this area of study, valuable advice and encouragement. I would also like to thank Arkady Zurkov, Dmitry Ryabogin, Alexander Fish and Professor Daniel Wise for helpful conversations, and the referee for helpful remarks.

2. Preliminary remarks

2.1. Hyperbolic metric spaces in the sense of Gromov. Let X be a path-connected metric space with distance function d. For an isometry i: [0, α] ⟶ X, the image p of I is a path in X, and we call α the length of p, denoted by l(p). X is a geodesic metric space if for all points x, y ∈ X there is an isometric map i: [0, d(x, y)] ⟶ [xy] from the interval [0, d(x, y)] to a path [xy] in X joining x and y; the path [xy] between the point x and y realising the distance d(x, y), is a geodesic segment.

Definition 2.1. [8, 1.1.1](cf. [11, 1.1]) Given a base point w ∈ X, we define an inner product on X by

(x.y)_{w} = \frac{1}{2}(d(x, w) + d(w, y) − d(x, y)).

If there is a constant δ ≥ 0 such that

∀x, y, z ∈ X, (x.y)_{w} ≥ min{(x.z)_{w}, (z.y)_{w}} − δ,

we say that the pointed metric space (X, w) is δ−hyperbolic.

Proposition 2.2. [8, 1.1.2] If (X, w) is δ−hyperbolic then for each v ∈ X, (X, v) is 2δ−hyperbolic.

Definition 2.3. [8, 1.1.4] A metric space X is δ−hyperbolic if for each w ∈ X, (X, w) is δ−hyperbolic. One says that X is hyperbolic if there exists δ ≥ 0 such that (X, w) is δ−hyperbolic.

The following definition is attributed to Rips.

Definition 2.4. [1, 1.1.3] Given any three points x, y, z in X, we say that a triangle xyz of geodesics joining these points is δ−slim if for any point w on [xy] we have that

min{d(w, [xz]), d(w, [yz])} ≤ δ.

We say that triangles are slim in X if there is a constant δ such that all geodesic triangles in X are δ−slim.

Proposition 2.5. [8, 1.3.6] Let X be a geodesic metric space. (1) If X is δ−hyperbolic then all geodesic triangles in X are 4δ−slim.

(2) If all geodesic triangles in X are δ−slim then X is 8δ−hyperbolic.

It follows from Proposition 2.5 that a geodesic metric space X is hyperbolic if and only if geodesic triangles are slim in X.

2.2. Hyperbolic groups. Let G be a group, and let S be a finite set generating G. There is a natural surjection φ: F (S) ⟶ G where F (S) is the free group on S. We define the length l_{S}(g) of an element g of G with respect to the generators S to be the length of the shortest word in F (S) representing g :

l_{S}(g) = min{l(f) | f ∈ F (S), φ(f) = g}.

We define the distance between two elements g_{1}, g_{2} ∈ G in the following way: d_{S}(g_{1}, g_{2}) = l_{S}(g_{1}^{−1}g_{2}). This is called the word metric on G with

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 5

respect to the generating set S. We construct a geodesic metric space called the Cayley graph X(G, S) of G with respect to the generating set S. This graph has a vertex for each element of G and an oriented edge labelled s from g to gs for each g ∈ G and s ∈ S. The space is clearly arc-connected. The group G acts on X(G, S) by the multiplication on the left: the element g defines a map α_{g}, which maps a vertex h ∈ X(G, S) to the vertex gh. A metric is defined by assigning unit length to each edge, and defining the distance between two points to be the minimum length of paths joining them. With this metric, the left-action of G on X(G, S) is by isometries.

Definition 2.6. We say that a group G is word hyperbolic (which we abbreviate to hyperbolic) if it has a finite set S of generators such that the corresponding Cayley graph X(G, S) is a hyperbolic space.

According to [8, Theorem 4.3.1], the property of G to be a hyperbolic group does not depend on the chosen set of generators.

Finite groups, finitely generated free groups, fundamental groups of oriented manifolds (without boundary) of negative curvature, groups that have a presentation which satisfies the small cancellation condition C′(λ) for λ ≤ \frac{1}{6} (see [22] for the definition) are examples of hyperbolic groups.

Hyperbolic groups are necessarily finitely presented ([1, Theorem 2.2.12]). Subgroups of hyperbolic groups are not necessarily hyperbolic. Rips [30] gives an example of a hyperbolic group G that has a subgroup H which is not finitely presentable, hence, H is not hyperbolic.

The following theorem asserts that any infinite Abelian subgroup of a hyperbolic group is virtually cyclic. In particular, a cyclic subgroup of a hyperbolic group is necessarily embedded in a maximal cyclic subgroup so that a hyperbolic group does not contain locally cyclic, non-cyclic subgroups.

Theorem 2.7. [1, 3.3.5][8, 10.7.2] (cf. [26, Theorem 21.2]) Let G be a hyperbolic group, and let g ∈ G be an element of infinite order. Let C(g) denote the centraliser of g in G. Then C(g)/⟨g⟩ is finite.

In case G is torsion-free, each Abelian subgroup of G is cyclic.

2.3. Bestvina-Paulin method. We describe in brief a construction of a sequence of metric H−spaces converging to a real tree Y equipped

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with a non-trivial action of H. More details can be found in [3] and [28] where the whole approach was introduced for the first time, in [27] and in the expository paper [6].

We start with the following classical construction. Fix a compact metric space X and denote by C(X) the set of closed subsets of X. The open ϵ−neighborhood in X of A ⊂ X is denoted by V_{ϵ}(A).

Definition 2.8. [6, 1.1] The Hausdorff metric on C(X) is defined by:

D(A, B) = inf{ϵ | A ⊆ V_{ϵ}(B) and B ⊆ V_{ϵ}(A)}.

The notion of Hausdorff metric can be generalised to the case when there is no ambient space containing metric spaces A and B.

Definition 2.9. [6, 1.4] An ϵ−approximation between two metric spaces A and B is a subset R ⊆ A × B such that:

(1) the projections of R to A and to B are onto;

(2) if (x, y), (x′, y′) ∈ R then |d_{A}(x, x′) − d_{B}(y, y′)| < ϵ. If there exists an ϵ−approximation between A and B then we write A ∼_{ϵ} B. The Hausdorff-Gromov distance between A and B is:

D_{H}(A, B) = inf{ϵ | A ∼_{ϵ} B}.

If there exists no ϵ such that A ∼_{ϵ} B, then D_{H}(A, B) is infinite. We say that a sequence of metric spaces C_{n} converges to C in the Hausdorff-Gromov topology if and only if D_{H}(C_{n}, C) → 0 as n → ∞.

Lemma 2.10. [28, 1.1] For every pair A, B of metric spaces, we have D(A, B) = 2D_{H}(A, B).

Limits of sequences of spaces in the Hausdorff-Gromov topology are not unique in general (note that if A is a dense subset of B then D_{H}(A, B) = 0.) However, for compact metric spaces the situation is different.

Proposition 2.11. [6, 1.5] Two compact metric spaces A and B are isometric if and only if the Hausdorff-Gromov distance between them is zero.

Thus, in the case of compact spaces, limits are unique whenever they exist. The following compactness criterion was proved by Gromov in [12]. To state the criterion, we need one more definition.

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Definition 2.12. [6, 1.6] We say that a family {C_{i}}_{i∈I} of compact metric spaces is uniformly compact if there is a uniform bound on their diameters, and for every ϵ > 0 there exists an integer N(ϵ) such that each C_{i} can be covered by N(ϵ) balls of radius ϵ.

Theorem 2.13 (Gromov). If a sequence {C_{i}}_{i∈ℕ} of compact metric spaces is uniformly compact then there is a subsequence which converges in the Hausdorff-Gromov metric.

The notion of Hausdorff-Gromov convergence is to be extended, in order to be adapted to study non-compact spaces especially in case when limit spaces are not locally compact (for instance, real trees). A suitable notion (equivariant Gromov convergence) was introduced by Paulin [28] (see also, Bestvina [3]).

Definition 2.14. A sequence of G−spaces X_{m} converges for the Gromov topology to a G−space X_{∞} if for every ϵ > 0, for every finite subset P in G, and for every compact subset K in X_{∞}, there exists (for m big enough) a compact subset K_{m} in X_{m}, and an ϵ−approximation R_{m} between K and K_{m}, which is P −equivariant, i.e. ∀x ∈ K, ∀x′ ∈ K_{m}, ∀α ∈ P, αx ∈ K and xR_{m}x′ ⇒ αx′ ∈ K_{m} and αxR_{m}αx′.

The idea is that for every compact subset K of the limit space X_{∞} and for m big enough, we can find in X_{m} a compact subset K_{m} such that K and K_{m} have similar structures and the actions of a finite subset of G on these two compacts are similar. It should be noted that even in cases where G is trivial, equivariant Gromov convergence does not imply Hausdorff-Gromov convergence. With respect to equivariant Gromov convergence, limits are not unique in general. However, Paulin has shown that under suitable convexity hypotheses, the compactness criterion of Gromov (Theorem 2.13 above) can be extended. In this article we produce a real tree endowed with an action of a group H as a limit of δ−hyperbolic spaces. In order to understand algebraic properties of the group, we study the properties of the limit tree. These properties of the limit tree can be deduced by looking at Hausdorff-Gromov convergence on compact subsets. The construction of the limit tree is based on the following theorem.

Theorem 2.15 (Paulin). [27, 2.3] Let {X_{m}}_{m=1}^{∞} be a sequence of δ_{m}− hyperbolic spaces with δ_{∞} = lim δ_{m} < ∞. Let G be a countable group

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isometrically acting on X_{m}. Suppose there exists a base point x_{m} in X_{m} such that for every finite subset P of G, the sets of geodesics between the images of x_{m} under P form a sequence of totally bounded metric spaces. Then there is a subsequence converging in the Gromov topology to a δ_{∞}−hyperbolic space X_{∞} endowed with a left isometric action of G.

A proof of this theorem can be derived from Gromov’s compactness criterion and the following lemma.

Lemma 2.16. [27, 2.4] Let {X_{m}}_{m∈ℕ} be a sequence of δ_{m}−hyperbolic spaces, with δ_{∞} = lim δ_{m} < ∞. Let X_{∞} be a geodesic space such that for every compact subset K in X_{∞} and every ϵ > 0, there exists (for m big enough) a compact subset K_{m} in X_{m}, and an ϵ−approximation R_{m} between K and K_{m}. Then X_{∞} is a δ_{∞}−hyperbolic space.

2.4. Real trees.

Definition 2.17. Let X be a metric space, and let x, y ∈ X. An arc from x to y is the image of a topological embedding σ: [a, b] ⟶ X of a closed interval [a, b] such that σ(a) = x and σ(b) = y. We say that X is a real tree if for any x, y ∈ X there is a unique arc from x to y and this arc is a geodesic segment.

It turns out that real trees are 0−hyperbolic spaces precisely.

Theorem 2.18. [8, 1.2.2, 3.4.1] A geodesic metric space X is 0−hyperbolic if and only if X is a real tree.

2.5. Rips’ theory of groups acting on real trees. Let a group G act on a real tree Y by isometries. An element g ∈ G is elliptic if g fixes a point of Y. The action of G on Y is trivial if some point of Y is fixed by the whole group. Otherwise, the action is nontrivial. A subtree T of Y is nondegenerate if T contains more than one point. A nondegenerate subtree T of Y is stable if for every nondegenerate subtree T′ of T, we have stab(T′) = stab(T). A nontrivial action is stable if every nondegenerate subtree of Y contains a stable subtree.

Roughly speaking, Rips’ theory is a “machine” that allows one to put information on a stable isometric action of a finitely presentable group G on a real tree Y into canonical form. Namely, the output of

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 9

the “Rips machine” is a graph of groups decomposition of G which is canonical with respect to the given action. Sela generalized Rips’ theory to finitely generated groups ([35] and [36]). In [35] the real tree Y is divided into (canonical) distinct components; vertex groups in the graph of groups are conjugates of the stabilizers of these components, and a vertex group (which is a subgroup of G) acts on the corresponding component according to one of several canonical types of actions.

The discussion of Rips’ theory can be found in the papers of Bestvina and Feighn [5], Rips [32], Paulin [29], Gaboriau, Levitt and Paulin [14], Guirardel [13]; see also the survey of Bestvina [4] and the book of M. Kapovich [18, Chapter 14]. One of the most important consequences of Rips’ theory is the following theorem.

Theorem 2.19 (Bestvina and Feighn). [5, Theorem 9.5] Let G be a finitely presented group with a nontrivial, stable and minimal action on a tree T. Then either

(1) G splits over an extension E−by-cyclic where E fixes an arc of T, or

(2) T is a line. In this case, G splits over an extension of the kernel of the action by a finitely generated free Abelian group.

Paulin [27] proved that if G is hyperbolic and has an infinite outer automorphism group then G admits a nontrivial, stable action on a real tree Y, with virtually cyclic segment stabilizers. Therefore we have the following property of hyperbolic groups.

Proposition 2.20. [5, Corollary 1.3], [4, Theorem 7.3] If G is a word hyperbolic group and Out(G) is infinite, then G splits over a virtually cyclic subgroup.

Notice that if in Proposition 2.20 G is torsion-free then it splits over a cyclic group. An important observation is that this property fails for subgroups of hyperbolic groups. Consider the following example. Apply the Rips construction [30, Theorem] to the infinite cyclic group ℤ = ⟨t⟩ and obtain a short exact sequence of groups:

1 ⟶ H ⟶ G ⟶ ℤ ⟶ 1

where G that has the presentation:

G = ⟨t, b_{1}, b_{2} | t^{−1}b_{1}tb_{1}b_{2}^{p_{1}}b_{1}b_{2}^{p_{2}} … b_{1}b_{2}^{p_{m}}, t^{−1}b_{2}tb_{1}b_{2}^{q_{1}}b_{1}b_{2}^{q_{2}} … b_{1}b_{2}^{q_{m}}⟩

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is torsion-free hyperbolic.

First, note that the normal subgroup H = ⟨b_{1}, b_{2}⟩ of G is free. Indeed, consider the free group F_{2} = ⟨b_{1}, b_{2}⟩ of rank two and its subgroup A = ⟨(b_{1}b_{2}^{p_{1}}b_{1}b_{2}^{p_{2}} … b_{1}b_{2}^{p_{m}})^{−1}, (b_{1}b_{2}^{q_{1}}b_{1}b_{2}^{q_{2}} … b_{1}b_{2}^{q_{m}})^{−1}⟩. There is an isomorphism φ: F_{2} ⟶ A where φ(b_{1}) = (b_{1}b_{2}^{p_{1}}b_{1}b_{2}^{p_{2}} … b_{1}b_{2}^{p_{m}})^{−1} and φ(b_{2}) = (b_{1}b_{2}^{q_{1}}b_{1}b_{2}^{q_{2}} … b_{1}b_{2}^{q_{m}})^{−1}. Now,

G ≅ F_{2}∗_{⟨t⟩} = ⟨t, b_{1}, b_{2} | t^{−1}b_{1}t = φ(b_{1}), t^{−1}b_{2}t = φ(b_{2})⟩

hence by a theorem proved by Higman, H. Neumann and B. Neumann in [17] (see also [19, Chapter X, §38.6]), F_{2} ≅ H.

Furthermore, since ℤ embeds into Out(H) (otherwise, we have a free Abelian group of rank two embedded into the hyperbolic group G, which is a contradiction), the outer automorphism group Out(H) of H is infinite. Now, we follow [21] and add to the presentation of H relations that force H to have the trivial abelianization. According to [21, Lemma 5.2], the factor group Ĥ of H does not split over a cyclic group.

2.6. The canonical JSJ decomposition of a group. The theory of canonical JSJ decompositions (based on the Rips’ theory of groups acting on real trees) was introduced first by Sela [37] for (word) hyperbolic groups. It was generalized by Rips and Sela [34] to finitely presented groups. A version of this theory appears in [36] where the JSJ decomposition is constructed for a finitely generated, non-finitely presentable group using Acylindrical accessibility (see [35] for the definition). The canonical JSJ decomposition of a group G is a canonical decomposition of G into a graph of groups which “contains” (in an appropriate sense) the information on all small splittings of G. The expression “small splitting” means that the edge stabilisers in the canonical graph of groups are groups of some special class (for instance, in the JSJ decomposition of a hyperbolic group edge stabilisers are virtually cyclic). We refer the reader to [31] and [34] for the definitions and detailed explanations. It should be mentioned that (different) theories of JSJ decompositions were developed by Dunwoody and Sageev [9] and Fujiwara and Papasoglu [10] who used different approaches. These theories enable one to consider splittings of finitely presented groups over subgroups taken from wider classes of groups.

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3. General case

We start with the construction of [36, Section 1]. If ψ has a nontrivial kernel then the epimorphisms ψ^{m} have different kernels. Indeed, it is obvious that ∀m, ker(ψ^{m}) ⊆ ker(ψ^{m+1}). Now, consider the following commutative diagram:

H/ker(ψ^{m}) ⟶^{ψ^{m}} H
ψ̄↓  ↓^{ψ}
H/ker(ψ^{m}) ⟶^{ψ^{m}} H

Since ψ is an epimorphism, the map ψ^{m}: H/ker(ψ^{m}) ⟶ H is an isomorphism. The map ψ̄: H/ker(ψ^{m}) ⟶ H/ker(ψ^{m}) is defined by ψ̄(h·ker(ψ^{m})) = ψ(h)·ker(ψ^{m}) so that ker(ψ̄) = ker(ψ^{m+1})/ker(ψ^{m}). Now, suppose that for some n > m, ker(ψ^{n}) = ker(ψ^{m}). It follows that ker(ψ^{m}) = ker(ψ^{m+1}). Therefore, ψ̄ is an isomorphism, and from the above diagram, it follows that ψ is an isomorphism, which is a contradiction to our assumption. Hence, for n ≠ m, ∀g ∈ Γ, ψ^{n} ≠ ĝ ∘ ψ^{m} (ĝ denotes the internal automorphism of Γ induced by the element g), for if ∀h ∈ H, ψ^{n}(h) = gψ^{m}(h)g^{−1} then ker(ψ^{n}) = ker(ψ^{m}). Let X be the Cayley graph of the ambient hyperbolic group Γ with respect to some finite generating set, and let α_{1}, …, α_{t} be generators of H. By definition, X is a δ−hyperbolic space. For each m we pick an element γ_{m} ∈ Γ where the (integer-valued) function

f_{m}(γ) = max_{1≤i≤t} d(id, γψ^{m}(α_{i})γ^{−1})

achieves its minimum, and set μ_{m} to be μ_{m} = f_{m}(γ_{m}) = min_{γ∈Γ} f_{m}(γ). If {μ_{m}}_{m=1}^{∞} has a bounded subsequence then by pigeon-hole argument there exist n ≠ m and γ ∈ Γ such that ψ^{n}(α_{i}) = γψ^{m}(α_{i})γ^{−1} for all 1 ≤ i ≤ t which is a contradiction; hence, μ_{m} ⟶ ∞. We set {(X_{m}, x_{m})}_{m=1}^{∞} to be the sequence of pointed metric spaces obtained by rescaling the metric on (X, id) by \frac{1}{μ_{m}}; x_{m} = γ_{m}^{−1} · id. The subgroup H acts isometrically on (X_{m}, x_{m}) by left multiplication twisted by the homomorphisms γ̂_{m} ∘ψ^{m}(h). We mean, there is an equivariant diagram

H × X_{m} ⟶ X_{m}
(γ̂_{m}∘ψ^{m},id)↓  ↓
Γ × X_{m} ⟶ X_{m}

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The spaces {(X_{m}, x_{m})}_{m=1}^{∞} endowed with the left isometric action of H, satisfy the assumptions of Theorem 2.15. Indeed, X_{m} is a δ_{m}−hyperbolic space for δ_{m} = \frac{δ}{μ_{m}} so that lim δ_{m} = 0, hence, the limit space is a real tree. Now, let P be a finite subset of H, every element h ∈ P is a word in the generators α_{1}, …, α_{t} of H; let L_{P} denote the maximal length of these words: L_{P} = max_{h∈P} |h|. Using the triangle inequality, we show that the maximal displacement of x_{m} in X_{m} under the action of the elements of P is bounded by a constant independent on m :

max_{h∈P} \frac{1}{μ_{m}} d(x_{m}, ψ^{m}(h) · x_{m}) ≤ \frac{1}{μ_{m}} L_{P} max_{1≤i≤t} d(x_{m}, ψ^{m}(α_{i}) · x_{m}) = L_{P}

where the equality follows from the definition of μ_{m}. Therefore the diameter of the set of geodesics between the images of x_{m} under P is bounded by 2L_{P} for all m. Thus we get a subsequence of pointed hyperbolic metric spaces converging in the Hausdorff-Gromov topology to a pointed real tree (Y, y_{0}) endowed with an action of H. To save notation, we continue to denote the converging subsequence by {(X_{m}, x_{m})}_{m=1}^{∞}. By the construction, the action of H on Y is not trivial (a detailed proof can be found in [27]), and ∀h ∈ H, d(y_{0}, h · y_{0}) = lim_{m→∞}[\frac{1}{μ_{m}} d(x_{m}, ψ^{m}(h) · x_{m})].

With the action of H on the limit tree we associate a “group at infinity” H_{∞} = H/K_{∞} where K_{∞} = {h ∈ H | ∀y ∈ Y h(y) = y} is the kernel of the action of H on Y. Note that there is an action H_{∞} × Y ⟶ Y defined by (h · K_{∞})(y) = h(y). Let η: H ⟶ H_{∞} denote the natural homomorphism. According to [36, Lemma 1.4], K_{∞} = ∪_{m≥1}ker(ψ^{m}). Due to the structure of ker(η) = K_{∞}, we have the following connection between subgroups of H that have the same image under η.

Lemma 3.1. If K_{1} and K_{2} are finitely generated subgroups of H such that η(K_{1}) = η(K_{2}) then there exists an integer n such that ψ^{n}(K_{1}) = ψ^{n}(K_{2}).

Proof. Let K_{1} = ⟨k_{1}, …, k_{t}⟩. By the condition, there are elements h_{1}, …, h_{t} ∈ ker(η) such that k_{1}h_{1}, …, k_{t}h_{t} ∈ K_{2}. Since ker(η) = ∪_{m≥1}ker(ψ^{m}), there exists u ≥ 1 such that ψ^{u}(h_{i}) = 1, i = 1, …, t.

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Then

ψ^{u}(K_{1}) =ψ^{u}(⟨k_{1}, …, k_{t}⟩) = ⟨ψ^{u}(k_{1}), …, ψ^{u}(k_{t})⟩ =
⟨ψ^{u}(k_{1}h_{1}), …, ψ^{u}(k_{t}h_{t})⟩ ⊆ ψ^{u}(K_{2}).

Similarly, there exists v ≥ 1 such that ψ^{v}(K_{2}) ⊆ ψ^{v}(K_{1}). Then for n = max{u, v} we have ψ^{n}(K_{1}) = ψ^{n}(K_{2}), as required. □

Furthermore, we can define a map ψ_{∞}: H_{∞} ⟶ H_{∞} as follows. For each x ∈ H_{∞} let h_{x} ∈ H satisfy η(h_{x}) = x, then ψ_{∞}(x) = η(ψ(h_{x})). According to [36, Lemma1.10], ψ_{∞} is an automorphism of H_{∞}. We have the following diagram

H ⟶^{ψ} H
η↓  ↓^{η}
H_{∞} ⟶^{ψ_{∞}} H_{∞}

Further, we follow [36, Section 3]. As H_{∞} is finitely generated and not finitely presented ([36, Lemma 1.4]), H_{∞} is not a free group, and H_{∞} = H_{∞}^{1} ∗⋯∗H_{∞}^{l} ∗F_{r} for some finitely generated non-cyclic, torsionfree, freely indecomposable groups H_{∞}^{1}, …, H_{∞}^{l} and a possible additional free factor F_{r}.

Claim 3.2. With the notation above, there is an integer k and an element x_{i} in H_{∞} such that (ψ_{∞})^{k}(H_{∞}^{i}) = x_{i}H_{∞}^{i}x_{i}^{−1}, i = 1, …, l.

Proof. Since the inherited map ψ_{∞}: H_{∞} ⟶ H_{∞} is an automorphism, H_{∞} = ψ_{∞}(H_{∞}) = ψ_{∞}(H_{∞}^{1})∗⋯∗ψ_{∞}(H_{∞}^{l})∗ψ_{∞}(F_{r}). Since both H_{∞}^{i} and ψ_{∞}(H_{∞}^{i}) are freely indecomposable for all i = 1, 2, …, l, by Kurosh’s theorem we have the following:

∃a_{i} ∈ H_{∞} : ψ_{∞}(H_{∞}^{i}) ⊆ a_{i}H_{∞}^{τ(i)}a_{i}^{−1} i = 1, 2, …, l  (1)
∃b_{j} ∈ H_{∞} : H_{∞}^{j} ⊆ b_{j}ψ_{∞}(H_{∞}^{ρ(j)})b_{j}^{−1} j = 1, 2, …, l.  (2)

Moreover, for all x ∈ H_{∞}, if i ≠ j then H_{∞}^{i} ∩ xH_{∞}^{j}x^{−1} = {1}. Since

ψ_{∞}(H_{∞}^{i}) ⊆ a_{i}H_{∞}^{τ(i)}a_{i}^{−1} ⊆ a_{i}b_{τ(i)}ψ_{∞}(H_{∞}^{ρ(τ(i))})b_{τ(i)}^{−1}a_{i}^{−1},

ρ(τ(i)) = i for all i = 1, 2, …, l. Similarly,

H_{∞}^{j} ⊆ b_{j}ψ_{∞}(H_{∞}^{ρ(j)})b_{j}^{−1} ⊆ b_{j}a_{ρ(j)}H_{∞}^{τ(ρ(j))}a_{ρ(j)}^{−1}b_{j}^{−1}

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so that τ(ρ(j)) = j for all j = 1, 2, …, l. Hence ρ, τ ∈ Sym{1, 2, …, l}, in particular ∃k such that τ^{k} = id_{{1,2,...,l}}. (1) implies that

(ψ_{∞})^{k}(H_{∞}^{i}) ⊆ (ψ_{∞})^{k−1}(a_{i})(ψ_{∞})^{k−2}(a_{τ(i)}) … a_{τ^{k−1}(i)}H_{∞}^{τ^{k}(i)}
a_{τ^{k−1}(i)}^{−1} … (ψ_{∞})^{k−2}(a_{τ(i)})^{−1}(ψ_{∞})^{k−1}(a_{i})^{−1} = x_{i}H_{∞}^{i}x_{i}^{−1}.

On the other hand,

⟨x_{1}H_{∞}^{1}x_{1}^{−1}, x_{2}H_{∞}^{2}x_{2}^{−1}, …, x_{l}H_{∞}^{l}x_{l}^{−1}⟩ =
x_{1}H_{∞}^{1}x_{1}^{−1} ∗ x_{2}H_{∞}^{2}x_{2}^{−1} ∗ ⋯ ∗ x_{l}H_{∞}^{l}x_{l}^{−1} ⊆ H_{∞} = (ψ_{∞})^{k}(H_{∞}) =
(ψ_{∞})^{k}(H_{∞}^{1}) ∗ (ψ_{∞})^{k}(H_{∞}^{2}) ∗ ⋯ ∗ (ψ_{∞})^{k}(H_{∞}^{l}) ∗ (ψ_{∞})^{k}(F_{r}),

and the claim follows. □

Lemma 3.3. With the notation above, there exist subgroups H_{1}, H_{2}, …, H_{l} of H which satisfy the following properties (i = 1, …, l) :

∃k ∈ ℤ : ∀i ∃g_{i} ∈ H such that ψ^{k}(H_{i}) = g_{i}H_{i}g_{i}^{−1};  (3)
∃m ∈ ℤ : ∀i η(H_{i}) = (ψ_{∞})^{m}(H_{∞}^{i}).  (4)

We can also find a free subgroup M of rank r in H such that η maps M isomorphically onto (ψ_{∞})^{m}(F_{r}).

Proof. Since H_{∞}^{i} is f.g. (which follows from Grushko’s theorem), we can find a f.g. subgroup H^{i} of H such that H_{∞}^{i} = η(H^{i}), i = 1, 2, …, l. By Claim 3.2, (ψ_{∞})^{k}(H_{∞}^{i}) = x_{i}H_{∞}^{i}x_{i}^{−1} for some integer k and x_{i} ∈ H_{∞}. Let h_{i} ∈ H be such that η(h_{i}) = x_{i}. We have

η(ψ^{k}(H^{i})) = (ψ_{∞})^{k}(H_{∞}^{i}) = x_{i}H_{∞}^{i}x_{i}^{−1} = η(h_{i}H^{i}h_{i}^{−1}).

Then by Lemma 3.1, there is n_{i} such that ψ^{n_{i}}(ψ^{k}(H^{i})) = ψ^{n_{i}}(h_{i}H^{i}h_{i}^{−1}), i = 1, 2, …, l. Let m = max_{1≤i≤l}{n_{i}}. It follows that

ψ^{k}(ψ^{m}(H^{i})) = ψ^{m}(ψ^{k}(H^{i})) = ψ^{m}(h_{i}H^{i}h_{i}^{−1}) = ψ^{m}(h_{i})ψ^{m}(H^{i})ψ^{m}(h_{i})^{−1}.

Let H_{i} = ψ^{m}(H^{i}), g_{i} = ψ^{m}(h_{i}). Then ψ^{k}(H_{i}) = g_{i}H_{i}g_{i}^{−1}, and

η(H_{i}) = η(ψ^{m}(H^{i})) = (ψ_{∞})^{m}(η(H^{i})) = (ψ_{∞})^{m}(H_{∞}^{i}).

Furthermore, there exist elements f_{1}, …, f_{r} ∈ H such that (ψ_{∞})^{m}(F_{r}) = ⟨η(f_{1}), …, η(f_{r})⟩; the subgroup M = ⟨f_{1}, …, f_{r}⟩ of H is free since η maps M onto the free group F_{r} of the same rank, and the hopficity of free groups implies that η restricted to M, η|_{M}: M ⟶ (ψ_{∞})^{m}(F_{r}) is an isomorphism. □

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 15

Corollary 3.4. There is an integer number n such that

ψ^{n}(H) = ⟨ψ^{n}(H_{1}), …, ψ^{n}(H_{l}), ψ^{n}(M)⟩.

Proof. With the notation of Lemma 3.3, let L_{∞}^{i} = η(H_{i}) = (ψ_{∞})^{m}(H_{∞}^{i}), (i = 1, 2, …, l), and let M_{∞} = η(M) = (ψ_{∞})^{m}(F_{r}), so that

H_{∞} = L_{∞}^{1} ∗ ⋯ ∗ L_{∞}^{l} ∗ M_{∞}.  (5)

Note that for each i = 1, 2, …, l, L_{∞}^{i} is a finitely generated, non-cyclic, freely indecomposable subgroup. Since by Lemma 3.3 η(H) = H_{∞} = η(⟨H_{1}, …, H_{l}, M⟩), by Lemma 3.1 for some n

ψ^{n}(H) = ψ^{n}(⟨H_{1}, …, H_{l}, M⟩) = ⟨ψ^{n}(H_{1}), …, ψ^{n}(H_{l}), ψ^{n}(M)⟩,

as claimed. □

Our aim is to prove the following theorem.

Theorem 3.5. With the notation above, η: H ⟶ H_{∞} is an isomorphism.

The proof of Theorem 3.5 is derived from the following important particular case.

Theorem 4.11. If H_{∞} is freely indecomposable then η: H ⟶ H_{∞} is an isomorphism.

(The proof of Theorem 4.11 is the content of Section 4.)

Proof of Theorem 3.5. First, we show that

η|_{ψ^{n}(H_{i})} i = 1, 2, …, l,  is an isomorphism.  (6)

In order to prove (6), in the statement of Theorem 4.11 we replace the subgroup H by ψ^{n}(H_{i}); by η in the statement of Theorem 4.11 we mean the map corresponding to the epimorphism φ: ψ^{n}(H_{i}) ⟶ ψ^{n}(H_{i}) where φ(a) = ψ^{n}(g_{i})^{−1}ψ^{k}(a)ψ^{n}(g_{i}) for a ∈ ψ^{n}(H_{i}). The map φ is well defined. Indeed, since by Lemma 3.3 ψ^{k}(H_{i}) = g_{i}H_{i}g_{i}^{−1}, we have ψ^{k}(ψ^{n}(H_{i})) = ψ^{n}(ψ^{k}(H_{i})) = ψ^{n}(g_{i}H_{i}g_{i}^{−1}) = ψ^{n}(g_{i})ψ^{n}(H_{i})ψ^{n}(g_{i})^{−1}. Now, we claim that the “group at infinity” H_{∞} is isomorphic to the image (ψ_{∞})^{n}(L_{∞}^{i}) of ψ^{n}(H_{i}). (Then H_{∞} is freely indecomposable, so the conditions of Theorem 4.11 are satisfied.) By the definition of L_{∞}^{i} (see the proof of Corollary 3.4) (ψ_{∞})^{n}(L_{∞}^{i}) = η(ψ^{n}(H_{i})) ≅ ψ^{n}(H_{i})/(K_{∞} ∩ ψ^{n}(H_{i})). On the other hand, by our construction ([36, Section 1], cf. the beginning of this Section) H_{∞} = ψ^{n}(H_{i})/N_{∞} where N_{∞} denotes

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the kernel of the action of ψ^{n}(H_{i}) on a limit tree T. Therefore, in order to prove our claim it is enough to show that

N_{∞} = K_{∞} ∩ ψ^{n}(H_{i})  (7)

Denote g = ψ^{n}(g_{i}), hence φ = ĝ^{−1} ∘ ψ^{k}. It is not difficult to see that for every positive integer u and a ∈ ψ^{n}(H_{i}), the following holds:

\begin{matrix}φ^{u}(a) = (g^{−1}ψ^{k}(g^{−1}) … ψ^{(u−1)k}(g^{−1}))ψ^{uk}(a)(ψ^{(u−1)k}(g) … ψ^{k}(g)g) \\ = h^{−1}ψ^{uk}(a)h. \end{matrix}(8)

Equality (8) implies that ker(φ^{u}) = ker(ψ^{uk}) ∩ ψ^{n}(H_{i}). Recall that by [36, Lemma 1.4], K_{∞} = ∪_{u≥1}ker(ψ^{u}) and N_{∞} = ∪_{u≥1}ker(φ^{u}). Also we

have ∪_{u≥1}ker(ψ^{uk}) = ∪_{u≥1}ker(ψ^{u}). Therefore

K_{∞} ∩ ψ^{n}(H_{i}) = ∪_{u≥1}(ker(ψ^{uk}) ∩ ψ^{n}(H_{i})) = ∪_{u≥1}ker(φ^{u}) = N_{∞}

and we get (7). (In particular, if ker(φ) is trivial then K_{∞} ∩ ψ^{n}(H_{i}) is trivial as well, and (6) follows.) Therefore, the conditions of Theorem 4.11 are satisfied. It follows that η: ψ^{n}(H_{i}) ⟶ (ψ_{∞})^{n}(L_{∞}^{i}) is an isomorphism, and we get assertion (6).

Now, the assertion (6) and the existence of normal forms relative to the free decomposition (5) of H_{∞} imply that

\begin{matrix}⟨ & ψ^{n}(H_{1}), …, ψ^{n}(H_{l}), ψ^{n}(M)⟩ ≅ ψ^{n}(H_{1}) ∗ ⋯ ∗ ψ^{n}(H_{l}) ∗ ψ^{n}(M) \\ ≅ (ψ_{∞})^{n}(L_{∞}^{1}) ∗ ⋯ ∗ (ψ_{∞})^{n}(L_{∞}^{l}) ∗ (ψ_{∞})^{n}(M_{∞}) & (9) \end{matrix}

Then in view of Corollary 3.4 and (5), η: ψ^{n}(H) ⟶ (ψ_{∞})^{n}(H_{∞}) is an isomorphism, which implies the assertion of Theorem 3.5. Thus, we have obtained a contradiction to our assumption that H_{∞} is a proper factor of H. Therefore, ker(ψ) is necessarily trivial, so that H is Hopf. This proves our main theorem.

Theorem 3.5. Finitely generated subgroups of a torsion-free hyperbolic group are Hopf.

4. A freely indecomposable subgroup

In this section, Γ denotes a torsion-free hyperbolic group, H ⊂ Γ is a f.g. subgroup of Γ and ψ: H ⟶ H is an epimorphism. We assume that the corresponding group at infinity H_{∞} is freely indecomposable. Our aim is to show that the natural homomorphism η: H ⟶ H_{∞} is

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 17

an isomorphism. From this point we assume familiarity with the basic notions of JSJ theory ([34], [36], [35]). We use the construction of the JSJ decomposition of H_{∞} which is the content of [36, Section 2]. The properties of the JSJ decomposition are listed in [36, Theorem 2.8]. We will use the following corollary of this Theorem.

Corollary 4.1. [36, Corollary 2.9] Suppose H_{∞} is freely indecomposable. Then the automorphism ψ_{∞} permutes the conjugacy classes of vertex and edge groups in the JSJ decomposition of H_{∞}. Furthermore, if E is a maximal cyclic edge group in the JSJ decomposition of H_{∞} and e ∈ E, then the conjugacy class of e is periodic under the automorphism ψ_{∞}.

According to [36, Theorem 1.7 and Proposition 1.9], the edge groups in the canonical JSJ decomposition of H_{∞} are maximal locally infinitecyclic subgroups. We obtain a new graph of groups decomposition of H_{∞} by collapsing all edges with non-cyclic stabilizers. We denote this new graph of groups by G((V, ℤ), T); edge stabilizers in this graph of groups are all infinite cyclic. We distinguish the following three types of vertex groups in G((V, ℤ), T) :

(1) a socket of a CMQ subgroup of H_{∞} (we refer the reader to [34, Section 5] for the definition);

(2) a group that splits over a locally infinite-cyclic, non-cyclic group;

(3) the other groups which we call regular vertex groups. We show that a vertex group of either of these types can be “lifted” isomorphically to a subgroup of H; arguments for vertex groups of different types are different (Lemma 4.2, Theorem 4.3 and Corollary 4.9). Then we conclude (Theorem 4.11) that H_{∞} is isomorphic to H. We follow [36, Section 3].

Let V_{∞}^{1}, …, V_{∞}^{n} be the vertex groups in G((V, ℤ), T). Since V_{∞}^{1}, …, V_{∞}^{n} are vertex groups in a Bass-Serre graph of groups where edge groups are cyclic and the fundamental group H_{∞} of the whole graph is finitely generated, the structure of the fundamental group of a graph of groups [38, Chapter I.5] implies that each of the vertex groups V_{∞}^{i}, i = 1, 2, …, n is finitely generated also.

Consider an arbitrary vertex group V_{∞} in G((V, ℤ), T). Let v_{1}, v_{2}, …, v_{s} ∈ H be elements for which V_{∞} = ⟨η(v_{1}), η(v_{2}), …, η(v_{s})⟩. Let V be

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the subgroup of H generated by these elements: V = ⟨v_{1}, v_{2}, …, v_{s}⟩. Our aim is to show that there is a positive integer m such that the natural homomorphism η puts ψ^{m}(V) isomorphically onto V_{∞}. For the vertex groups of the first type, we have the following lemma.

Lemma 4.2. Suppose, V_{∞} is a socket of a CMQ vertex group in the JSJ decomposition of H_{∞}. Then for some integer p, η maps ψ^{p}(V) isomorphically onto ψ_{∞}(V_{∞}).

Proof. By [37], V_{∞} is finitely presented hence in view of [36, Lemma 1.4], for V such that η(V) = V_{∞} the intersection of V with the kernel of η is a finite union: V ∩ ker(η) = ∪_{k=1}^{p}(V ∩ ker(ψ^{k})) which implies the claim. □

The following theorem asserts that in G((V, ℤ), T), there are no groups of the second type which amounts to saying that edge groups in the canonical JSJ decomposition of H_{∞} are all cyclic.

Theorem 4.3. The JSJ decomposition of H_{∞} does not contain any locally-cyclic, non-cyclic edge groups.

Proof. Identical with the proof of Theorem 3.2 of [36]. □

At this point, we are left with a regular vertex group V_{∞} in the JSJ decomposition of H_{∞} where all the edge groups are infinite cyclic. We show that necessarily ψ^{u}|_{V} = â ∘ ψ^{m}|_{V} for some integers u > m and a ∈ Γ (Theorem 4.8), and conclude that η maps ψ^{m}(V) isomorphically onto V_{∞} (Corollary 4.9).

First, we prove the following.

Claim 4.4. Let E = ⟨e⟩ ⊂ V_{∞} be a maximal cyclic edge group in the JSJ decomposition of H_{∞}, and let h_{e} ∈ H be an element for which η(h_{e}) = e. Then there exist integers u > m and an element g ∈ H such that

ψ^{u}(h_{e}) = gψ^{m}(h_{e})g^{−1}.  (10)

Proof. By [36, Corollary 2.9], the conjugacy class of e is periodic under the automorphism ψ_{∞}: H_{∞} ⟶ H_{∞}. This means that there are integers k > l and α ∈ H_{∞}^{i} such that ψ_{∞}^{k}(e) = αψ_{∞}^{l}(e)α^{−1}, which implies that η(ψ^{k}(c)) = αη(ψ^{l}(c))α^{−1} = η(h_{α}ψ^{l}(c)h_{α}^{−1}), where η(h_{α}) = α. Therefore, for some p we get ψ^{k+p}(c) = ψ^{p}(h_{α})ψ^{l+p}(c)ψ^{p}(h_{α}^{−1}) (cf.

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 19

Lemma 3.1), and for u = k + p, m = l + p and g = ψ^{p}(h_{α}), we obtain the claim. □

Claim 4.5. With the notation of Claim 4.4, there exists a finite set C = {c_{0}, c_{1}, …, c_{u−m−1}} of elements of H such that for each k ≥ m, ψ^{k}(h_{e}) is conjugated to c_{q} ∈ C where the index q, 0 ≤ q < u − m, depends on k.

Proof. The conclusion (10) of Claim 4.4 implies that

ψ^{u−m}(h_{e}) = β_{1}h_{e}β_{1}^{−1}τ_{1}

where β_{1} ∈ H, ψ^{m}(β_{1}) = g and τ_{1} ∈ ker(ψ^{m}). Now we iterate the epimorphism ψ^{u−m} :

ψ^{2(u−m)}(h_{e}) = ψ^{u−m}(ψ^{u−m}(h_{e})) = ψ^{u−m}(β_{1}h_{e}β_{1}^{−1}τ_{1})
= ψ^{u−m}(β_{1})(β_{1}h_{e}β_{1}^{−1}τ_{1})ψ^{u−m}(β_{1}^{−1})ψ^{u−m}(τ_{1})
= β_{2}h_{e}β_{2}^{−1}σ_{1}ψ^{u−m}(τ_{1}) = β_{2}h_{e}β_{2}^{−1}τ_{2}

for β_{2} = ψ^{u−m}(β_{1})β_{1}, σ_{1} = ψ^{u−m}(β_{1})τ_{1}ψ^{u−m}(β_{1}^{−1}) and τ_{2} = σ_{1}ψ^{u−m}(τ_{1}). As both σ_{1} and ψ^{u−m}(τ_{1}) are in ker(ψ^{m}), we have τ_{2} ∈ ker(ψ^{m}). Then for any integer p,

ψ^{p(u−m)}(h_{e}) = ψ^{u−m}(ψ^{(p−1)(u−m)}(h_{e})) = ψ^{u−m}(β_{p−1}h_{e}β_{p−1}^{−1}τ_{p−1})

= ψ^{u−m}(β_{p−1})β_{1}h_{e}β_{1}^{−1}τ_{1}ψ^{u−m}(β_{p−1}^{−1})ψ^{u−m}(τ_{p−1}) = β_{p}h_{e}β_{p}^{−1}τ_{p}

for some τ_{p} ∈ ker(ψ^{m}). Hence, ψ^{p(u−m)+m}(h_{e}) = ψ^{m}(β_{p})ψ^{m}(h_{e})ψ^{m}(β_{p}^{−1}). Furthermore, for any integer 0 < q < u − m, we have

ψ^{p(u−m)+q}(h_{e}) = ψ^{q}(ψ^{p(u−m)}(h_{e})) = ψ^{q}(β_{p}h_{e}β_{p}^{−1}τ_{p})
= ψ^{q}(β_{p})ψ^{q}(h_{e})ψ^{q}(β_{p}^{−1})ψ^{q}(τ_{p}),

so that ψ^{p(u−m)+q+m}(h_{e}) = ψ^{q+m}(β_{p})ψ^{q+m}(h_{e})ψ^{q+m}(β_{p}^{−1}). Thus for any k ≥ m, we obtain ψ^{k}(h_{e}) = γ_{p,q}c_{q}γ_{p,q}^{−1} where k = p(u − m) + q + m so that 0 ≤ q < u − m, and c_{q} = ψ^{q+m}(h_{e}). □

Assume that ∀a ∈ Γ, ψ^{u} ≠ â ∘ ψ^{m}, for u > m so that we can repeat the limit procedure described in Section 3 and (by rescaling the metric on the Cayley graph X of Γ), obtain a sequence of pointed metric V −spaces {(X_{k}, x_{k})}_{k=1}^{∞} converging to a real tree Y.

Lemma 4.6. With the notation above, let E = ⟨e⟩ ⊂ V_{∞} be a maximal cyclic edge group in the JSJ decomposition of H_{∞}, and let h_{e} ∈ V be

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such that η(h_{e}) = e. Then there is a sequence of points {x̃_{k}}_{k=1}^{∞} in X and a number L such that d(x̃_{k}, γ̂_{k} ∘ ψ^{k}(h_{e}) · x̃_{k}) ≤ L.

Proof. As h_{e} ∈ V, it can be expressed as a word in the generators of V : h_{e} = w(v_{1}, …, v_{s}). V is not necessarily invariant under the epimorphism ψ, but for each k, ψ^{k}(h_{e}) = w(ψ^{k}(v_{1}), …, ψ^{k}(v_{s})) ∈ H. In particular, the elements c_{q} = ψ^{q+m}(h_{e}), q = 1, 2, …, u − m − 1, (cf. Claim 4.5) can be expressed as words in the generators of H. Denote the maximal length of these words by

L = max_{0≤q<u−m} l(c_{q}).

Let γ_{k} be the element where the function

f_{k}(γ) = max_{1≤i≤s} d(id, γψ^{k}(v_{i})γ^{−1})

achieves its minimum: μ_{k} = f_{k}(γ_{k}) = min_{γ∈Γ} f_{k}(γ). (Cf. the definitions of γ_{m} and μ_{m} in Section 3). We set x̃_{k} = γ_{k}γ_{p,q} and using Claim 4.5 obtain

d(x̃_{k}, γ_{k}ψ^{k}(h_{e})γ_{k}^{−1} · x̃_{k}) = d(γ_{k}γ_{p,q}, γ_{k}γ_{p,q}c_{q}γ_{p,q}^{−1}γ_{k}^{−1} · γ_{k}γ_{p,q})
= d(id, c_{q} · id) ≤ L.
□

Corollary 4.7. With the assumptions of Lemma 4.6, e is elliptic when acting on the real tree Y.

Proof. The sequence of points {x̃_{k}} converges to a point in the tree Y which is fixed by h_{e} :

d_{k}(x̃_{k}, γ_{k}ψ^{k}(h_{e})γ_{k}^{−1} · x̃_{k}) = d_{k}(id, c_{q} · id) ≤ \frac{1}{μ_{k}} L ⟶_{k→∞} 0.

Therefore, e = η(h_{e}) is elliptic when acting on Y. □

Theorem 4.8. With the notation above, let V_{∞} be a regular vertex group in the JSJ decomposition of H_{∞}. Then there exist integers u > m and an element a ∈ Γ such that ψ^{u}(v_{j}) = aψ^{m}(v_{j})a^{−1} for all j = 1, 2, …, s.

Proof. We follow the proof of [36, Theorem 3.1] (the theorem in [36] is proved in case when ψ is an epimorphism of Γ).

Assume, ∀a ∈ Γ, ψ^{u} ≠ â ∘ ψ^{m}, for u > m. Then we can repeat the limit procedure described in Section 3 (cf. [36, Section 1]), and obtain

THE HOPF PROPERTY FOR SUBGROUPS OF HYPERBOLIC GROUPS 21

a stable action of V_{∞} on a real tree Y which satisfies all the properties listed in [36, Lemmas 1.3, 1.5 and 1.6]. Suppose V_{∞} admits a (nontrivial) free decomposition V_{∞} = A ∗ B in which every edge group E connected to the vertex stabilized by V_{∞} in the JSJ decomposition of H_{∞}, can be conjugated into either A or B. Then we can replace the vertex stabilized by V_{∞} with the edge of groups (where A and B are vertex groups and the edge stabilizer is trivial) so as to obtain a proper refinement of the JSJ decomposition of H_{∞}; in particular, it follows that H_{∞} is freely decomposable, a contradiction to our assumption. Hence, V_{∞} does not admit any non-trivial free decomposition in which every edge group E connected to the vertex stabilized by V_{∞} in the JSJ decomposition of H_{∞}, can be conjugated into one of the factors. Since every edge group E fixes a point in Y (Corollary 4.7), we can analyse the action of V_{∞} on Y using [36, Theorem 1.8]. This theorem implies that V_{∞} inherits a non-trivial splitting Δ over a locally cyclic subgroup from its action on the real tree Y. Corollary 4.7 implies that the splitting Δ of V_{∞} is compatible with the JSJ decomposition of H_{∞} which allows one to refine the JSJ decomposition of H_{∞} by replacing the vertex stabilised by V_{∞} with the graph of groups Δ. The new graph of groups is a proper refinement of the JSJ decomposition of H_{∞} since V_{∞} does not fix a vertex in it, and we get a contradiction with the canonical properties of the JSJ decomposition. The obtained contradiction completes the proof. □

Corollary 4.9. With the notation above, if V_{∞} is a regular vertex group in the JSJ decomposition of H_{∞} then for some m, the natural homomorphism η puts ψ^{m}(V) isomorphically onto V_{∞}.

This assertion follows from Theorem 4.8 and the following Claim.

Claim 4.10. With the notation above, if ψ^{u}(v_{j}) = aψ^{m}(v_{j})a^{−1}, j = 1, 2, …, s for some integers u > m and a ∈ Γ then the natural homomorphism η puts ψ^{m}(V) isomorphically onto V_{∞}.

Proof. We have in particular, that ker(ψ^{m}|_{V}) = ker(ψ^{u}|_{V}) so that ψ restricted to ψ^{m}(V) is an isomorphism, hence, η restricted to ψ^{m}(V) is an isomorphism as claimed. □

22 INNA BUMAGINA

At this point, each vertex group in the JSJ decomposition of H_{∞} can be “lifted” isomorphically to a subgroup of H. We conclude that H_{∞} is isomorphic to H.

Theorem 4.11. If H_{∞} is freely indecomposable then η: H ⟶ H_{∞} is an isomorphism.

Proof. Let V_{∞}^{i} and V_{∞}^{j} be two vertex groups in the JSJ decomposition of H_{∞}, let subgroups V^{i} and V^{j} be such that η(V^{i}) = V_{∞}^{i} and η(V^{j}) = V_{∞}^{j}, and let E = V_{∞}^{i} ∩ V_{∞}^{j} be the stabiliser of an edge joining the two vertices. In view of Theorem 4.3, we can assume that E is a maximal cyclic subgroup. Let E = ⟨e⟩; by [36, Corollary 2.9] (cf. Corollary 4.1), e has a periodic conjugacy class under ψ_{∞}: H_{∞} ⟶ H_{∞}, hence by Claim 4.4, for each h_{e} ∈ V^{i} ∩ V^{j} such that η(h_{e}) = e, ψ^{n}(h_{e}) = gψ^{m}(h_{e})g^{−1} for some n > m and g ∈ H_{∞}. Therefore by Claim 4.10, η maps E_{m} = ψ^{m}(V^{i}) ∩ ψ^{m}(V^{j}) isomorphically onto V_{∞}^{i} ∩ V_{∞}^{j}. According to Lemma 4.2, Theorem 4.3 and Corollary 4.9, there are integers m_{i} and m_{j} such that for k > max{m_{i}, m_{j}}, η maps ψ^{k}(V^{i}) and ψ^{k}(V^{j}) isomorphically onto V_{∞}^{i} and V_{∞}^{j} in correspondence. Then for k > max{m, m_{i}, m_{j}}, η maps ψ^{k}(V^{i}), ψ^{k}(V^{j}) and E_{k} isomorphically onto V_{∞}^{i}, V_{∞}^{j} and E in correspondence.

Let t_{∞}^{1}, …, t_{∞}^{b} be the Bass-Serre generators in the JSJ decomposition of H_{∞}, and let t^{1}, …, t^{b} ∈ H be such that η(t^{i}) = t_{∞}^{i} for i = 1, …, b. By [38, Chapter I.5], we have H_{∞} = ⟨V_{∞}^{1}, …, V_{∞}^{n}, t_{∞}^{1}, …, t_{∞}^{b}⟩.

Let α_{1}, …, α_{t} be a generating set for H. Then for j = 1, …, t, η(α_{j}) can be expressed as a word in the elements from the vertex subgroups V_{∞}^{i} and the elements t_{∞}^{i}. Since by [36, Lemma 1.4], K_{∞} = ∪_{m=1}^{∞}ker(ψ^{m}), it follows that there exists an integer k for which each generator α_{j} (j = 1, 2, …, t) of H can be written as α_{j} = w_{j}τ_{j} where τ_{j} ∈ ker(ψ^{k}), and w_{j} is a word in elements from the vertex subgroups V^{1}, V^{2}, …, V^{n} and in t^{1}, t^{2}, …, t^{b}. Hence, for each j = 1, 2, …, t, we have ψ^{k}(α_{j}) ∈ ⟨ψ^{k}(V^{1}), …, ψ^{k}(V^{n}), ψ^{k}(t^{1}), …, ψ^{k}(t^{b})⟩. Since ψ is an epimorphism, H is generated by the t−tuple ψ^{u}(α_{1}), …, ψ^{u}(α_{t}) for every integer u, so H = ⟨ψ^{k}(V^{1}), …, ψ^{k}(V^{n}), ψ^{k}(t^{1}), …, ψ^{k}(t^{b})⟩.

Now, we may increase k so that η maps ψ^{k}(V^{i}) isomorphically onto (ψ_{∞})^{k}(V_{∞}^{i}), and the intersections ψ^{k}(V^{i}) ∩ ψ^{k}(V^{j}) and ψ^{k}(t_{p}V^{j}t_{p}^{−1})∩ ψ^{k}(V^{i}) isomorphically onto the intersections (ψ_{∞})^{k}(V_{∞}^{i}) ∩ (ψ_{∞})^{k}(V_{∞}^{j}) and (ψ_{∞})^{k}(V_{∞}^{i}) ∩ (ψ_{∞})^{k}(η(t_{p})V_{∞}^{j}η(t_{p}))^{−1}, in correspondence. Finally, the existence of normal forms with respect to the JSJ decomposition of H_{∞} implies that η is an isomorphism from ψ^{k}(H) = H onto H_{∞}. □

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Institute of mathematics, the Hebrew University, Jerusalem 91904, Israel

  E-mail address: binna@math.huji.ac.il