Endomorphisms of Kleinian Groups.

Thomas Delzant and Leonid Potyagailo

1 Introduction.

In this paper we study cohopfian (or co-Hopf) property. Recall that a group G is cohopfian (or has the co-Hopf property) if any injective endomorphism f : G → G is surjective.

Answering a question of E. Rips, Z. Sela showed in [Se] that a torsion-free, non virtually cyclic word-hyperbolic group (in Gromov’s sense) is cohopfian if and only if it is not a non-trivial free product. Note that a non-trivial free product A ∗ B is never cohopfian, as it contains the proper subgroup A ∗ mBm^{−1} isomorphic to A ∗ B if m ∉ (A ∪ B). The cohopficity of 3-manifold groups was studied by many authors, we refer to the papers [PW] and [OP] where a more complete list of references on this subject is given.

If G splits as an HNN-extension, G = A∗_{C} = ⟨A, t | tCt^{−1} = φ(C)⟩, so that t centralizes C then G is not cohopfian (set f : G → G be the identity on A and f(t) = t^{2} ; then f is injective, not surjective). It was showed in [OP] that such example can be realized as a Kleinian group. Note that in this example the group G splits over a parabolic subgroup C which is of infinite index in the unique maximal parabolic subgroup C̃ of G containing C (where C̃ =< C, t >) so that C̃ is not conjugate into A. In such case we will refer to the group C and the corresponding splitting of G over C as essentially non-maximal. On the other hand it was also showed in [OP] that G is cohopfian provided it does not split over elementary subgroups at all. A natural question now is whether all non cohopfian torsion free one-ended Kleinian groups arise only in this way, in other words is G non cohopfian if and only if G has essentially non-maximal splittings over parabolic subgroups ? The main result of the paper (Theorem A below) is a criterion showing that essentially this is indeed the case.

Instead of studying directly the “absolute” cohopfian property of a group G, we extend this notion to the “relative” case. Let us fix a finite system E = {E_{1}, ...., E_{n}} of elementary subgroups of G (a “peripheral system”) and suppose that f : G → G is an endomorphism which sends each E_{i} into itself. Then Theorem B below guarantees that f : (G, E) → (G, E) is a surjective map of the pair (G, E) provided that G has no essentially non-maximal splittings over elementary subgroups relatively to the system E (i.e. every subgroup in E is elliptic in such splitting of G) and one more condition is satisfied.

This notion of the relative cohopfian property can be easily illustrated in the case of a surface with boundary. Suppose S is a compact surface of genus g > 1 whose boundary is a finite collection of loops α_{i} (i = 1, ...n). Let E_{i} be the cyclic peripheral subgroup of G = π_{1}(S) generated by α_{i} and E = {E_{1}, ..., E_{n}}. The group G is a free group and so is not cohopfian ; however it is cohopfian relatively to E, i.e. if f : (G, E) → (G, E) is an injective endomorphism sending each group E_{i} into itself then f is surjective.

The paper is organized as follows. Section 2 contains a preliminary material and the statements of the results. In Section 3 we collect some accessibility results for an (abstract) finitely presented group G equipped with a system of elementary subgroups C which we define there axiomatically. In particular, discrete elementary subgroups of Isom_{+}ℍ^{n} (i.e those whose limit set is a finite set) satisfy these axioms. Following the paper [De] we obtain a version of a finiteness theorem for acylindrical splittings of G originally due to Z. Sela [Se] by keeping track of the peripheral system E (E ⊂ C). We then use the method of the paper [De-Po] to construct a finite hierarchy of G over elementary subgroups relatively to the system E. These results will be applied to Kleinian groups in Section 4. There we show that up to automorphism a finitely presented Kleinian group G can admit at most finitely many splittings over elementary subgroups (Theorem D). Then for non cohopfian Kleinian groups we will obtain a relative version of the result [OP], namely: if the pair (G, E) is not cohopfian (i.e. there exists an injective non-surjective endomorphism f of (G, E)) then G splits non-trivially over elementary subgroups relatively to the system E (Proposition 4.7). Assuming the pair (G, E) to be not cohopfian we will then find in Section 5 another injective non surjective map F : (G, E) → (G, E) and a non-trivial splitting of G as a graph of groups such that F will send every vertex (resp. edge) group of it into itself. The non-triviality of this splitting invariant under F is the most delicate point here; this fact will be showed by induction on the length of a finite hierarchy of G. The existence of such splitting invariant under F will allow us to prove in Section 6 the sufficient condition of Theorem B.

The converse implication, i.e. if G admits an essentially non maximal splitting over a parabolic group, then G is not cohopfian, will be proven in Section 7. The proof is based on a sequence of Lemmas about virtually abelian groups.

Acknowledgments. We thank Misha Kapovich for helpful discussions and for a suggestion how to prove Proposition 4.7. The second author is deeply grateful to the Max-Planck Institute für Mathematik of Bonn for the hospitality, mathematical stimulation and support during his one-year stay at the MPI where he has been working on this paper.

2 Preliminaries and Formulations of the Results.

Let ℍ^{n} denote the real hyperbolic space of dimension n. A group G is called Kleinian if G is a discrete subgroup of the orientation preserving part of the isometry group Isom_{+}ℍ^{n} of ℍ^{n}. The limit set Λ(G) of G is the set of accumulation points of some (any) orbit G(z) (z ∈ ℍ^{n}).

Recall that a Kleinian group H ⊂ Isom_{+}ℍ^{n} is elementary if its limit set Λ(H) ⊂ S_{∞}^{n−1} is a finite set, and H is a finite elementary group if and only if Λ(H) = ∅. An infinite elementary group H is loxodromic or parabolic accordingly to whether the limit set Λ(H) contains two points or one. By Bieberbach’s theorems (see e.g. [Ra]) every elementary subgroup H of Isom_{+}ℍ^{n} is virtually abelian i.e. contains a maximal free abelian subgroup A of finite index. The rank of the group A is called rank of H. A loxodromic elementary group is always virtually cyclic and is 2-ended. A parabolic subgroup of rank bigger than one is a one-ended group.

Notation. If C is elementary infinite, it is contained in a unique maximal elementary subgroup of G. This subgroup will be denoted C̃ throughout the paper.

A finitely generated Kleinian group G is geometrically finite if there exists an ε > 0 so that the hyperbolic volume of an ε-neighborhood of C(Λ(G))/G is finite, where C(Λ(G)) ⊂ ℍ^{n} is the convex hull of the limit set of G (i.e. the smallest convex subset of ℍ^{n} invariant under the G-action) is finite.

We say that G splits as a graph of groups X_{∗} = (X, (C_{e})_{e∈X^{1}}, (G_{v})_{v∈X^{0}}) (where C_{e} and G_{v} denote respectively edge and vertex groups of the graph X) if G is isomorphic to the fundamental group π_{1}(X_{∗}) (in the sense of Serre [Se]). The Bass-Serre tree T is the universal cover (in the sens of Serre) of the graph X = T/G. When X has only one edge, we will say that G splits as an amalgam (resp. an HNN-extension) if X has two vertices (resp. one vertex). We will need the following definitions:

Definition 2.1. Let G act on a tree T. A subset H of G is said elliptic (respectively hyperbolic) on T if H fixes a point in T (respectively does not fix a point on T); if T is the Bass-Serre tree of a splitting of G as a graph of groups, H is elliptic if and only if it is conjugate into a vertex group of this graph. We say that G splits relatively to a family of subgroups (E_{1}, ...E_{n}), or that (G, E) splits as a graph of groups, if G splits as a graph of groups in which all the groups E_{i}’s are elliptic.

Definition 2.2. Suppose G splits as a graph of groups

G = π_{1}(X, C_{e}, G_{v})  (1)

so that the edge groups (i.e. the groups C_{e}’s) of this graph are elementary. We say that the edge stabilizer C_{e} is essentially non-maximal if it is of infinite index in the maximal elementary subgroup C̃_{e} ⊂ G containing C_{e} and C̃_{e} is not conjugate into one of the vertex groups of the splitting (1). In other words, C_{e} is essentially non-maximal if the group C̃_{e} is not elliptic in the splitting (1). The splitting (1) is called essentially non-maximal if there exists at least one such an edge. In the opposite case we call the splitting (1) essentially maximal.

It turns out that the property to split (or not to split) over essentially non-maximal subgroups is closely related to the co-Hopf property mentioned in the Introduction.

Theorem A. Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary, geometrically finite, one-ended Kleinian group without 2-torsion. Then G is cohopfian if and only if the following two conditions are satisfied:

1) G does not have essentially non-maximal splittings.

2) G does not split as an amalgam G = A ∗_{C} C̃, so that C̃ is a maximal elementary subgroup of G and the normal subgroup of C̃ generated by C is of infinite index in C̃.

■

Note that if C is a non-trivial essentially non-maximal subgroup of a Kleinian group G, then |C̃ : C| = ∞, and so C is a parabolic subgroup of G, and rank (C) < rank C̃.

Corollary 2.3. Suppose G is a non-elementary, geometrically finite, one-ended Kleinian group without 2-torsion. Suppose that every elementary subgroup C over which G splits has a finite index in the maximal elementary subgroup C̃ containing it, then G is cohopfian.

As explained in the Introduction, the proof of Theorem A involves the study of the relative case.

Definition 2.4. Let G be a group endowed with a family E = (E_{1}, ..., E_{n}) of elementary subgroups. An endomorphism of G is called endomorphism of the pair (G, E) if it sends each E_{i} into itself.

The pair (G, E) is called cohopfian, if any injective endomorphism of (G, E) is surjective. We say that the pair (G, E) is \underline{one ended} if G does not split over finite subgroups relatively to the system E.

Theorem B. Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary, geometrically finite, Kleinian group without 2-torsion and E = {E_{1}, ..., E_{k}} be a family of elementary subgroups of G. Suppose that the pair (G, E) is one-ended. Then (G, E) is cohopfian if the following two conditions are satisfied:

1) The pair (G, E) does not have essentially non-maximal splittings over elementary subgroups.

2) The pair (G, E) does not split as an amalgam G = A∗_{C} C̃, so that C̃ is a maximal elementary subgroup of G and the normal subgroup of C̃ generated by C is of infinite index in C̃. ■

Remark: If the family E in Theorem B is empty or all groups E_{i} are finite the sufficient condition of Theorem A follows from Theorem B.

The condition of the essential non-maximality of a splitting introduced above is related to the following acylindricity condition of splittings introduced by Z. Sela in the torsion free case and in the paper [De] in the general case:

Definition 2.5. Let G split as a graph of groups G = π_{1}(X) with elementary edge stabilizers and T be the Bass-Serre tree corresponding to this splitting.

a) \underline{Torsion free case:} The splitting (and the tree T) is called K-acylindrical if the stabilizer of each segment on T of the diameter at least K is trivial.

b) \underline{General case:} The G-tree T is called (K, Φ)-acylindrical if the stabilizer of each segment on T of the diameter at least K is a finite group^{1}.

If G splits as a graph of groups G = π_{1}(X) , one says that this splitting is (K, Φ)−acylindrical if the Bass-Serre tree - the universal cover of X - is the (K, Φ)−acylindrical.

■

The condition for a group G not to have essentially non-maximal splittings turns out to be equivalent to the property that all its splittings over elementary subgroups are (K, Φ)-acylindrical for some uniform constant K (see Lemma 4.1 in Section 4).

1Here Φ stands for “finite”.

3 Accessibility of Finitely Presented Groups

In this section we collect some results about different versions of the accessibility (acylindrical and strong) of abstract finitely presented groups. These results will be applied in the next section to geometrically finite Kleinian groups.

Throughout this section G will denote an abstract (not necessarily Kleinian) group. Recall that the Bass-Serre G-tree T̂ is called a resolution of the G-tree T if there exists G-equivariant simplicial map ρ : T̂ → T.

If T is a G-tree and φ : G → G is a monomorphism of G into itself let φ^{∗}T denote the G-tree defined as follows: as the set φ^{∗}T is T, but the action of G on T obtained from the previous one by composing with φ:

g_{φ^{∗}T} (x) = φ(g)_{T} (x).  (2)

The stabilizer of a vertex v (edge e) of the tree φ^{∗}T is equal to φ^{−1}(G_{v}) (respectively φ^{−1}(C_{e})) where G_{v} (respectively C_{e}) is the stabilizer of v (respectively e) on T. Following [BeF1] we call a G-tree irreducible if it is minimal (i.e. there is no proper invariant subtree) and if the label of every vertex of valence two properly contains the labels of both edges incident to it (if the two edges are distinct).

We will further consider decompositions of finitely presented groups over so called elementary subgroups which we now define axiomatically:

Definition 3.1. Let G be a finitely presented group and C a fixed family of subgroups of G. We call the family C and every element C ∈ C \underline{elementary} if the following axioms are satisfied:

(1) If C ∈ C then every subgroup and every conjugate of C is in C.

(2) Every infinite subgroup belonging to C is contained in a unique maximal subgroup C̃ so that C̃ ∈ C. The union of an ascending sequence of finite elementary groups is elementary.

(3) Every subgroup of C satisfies the following fixed-point condition: whenever C acts on a simplicial tree τ, C preserves a point in τ, or a point on its ideal boundary ∂τ or a pair of points on ∂τ (possibly permuting them).

(4) If C ∈ C is an infinite maximal elementary subgroup then its normalizer in G is contained in C, i.e. gCg^{−1} = C implies that g ∈ C for all g ∈ G.

Examples of elementary families are well-known in the geometry of negatively curved spaces. Namely, discrete subgroups of the hyperbolic space ℍ^{n} or, more generally Hadamard spaces with a pinched negative curvature, are elementary in the classical sense if their limit set is a finite set. In this case they are also elementary according to our axioms (1 − 4). Indeed, the properties (1), (2) and (4) are easy exercises, the only property which is non-trivial is the axiom (3) which follows from Margulis’ lemma saying that every such group is virtually nilpotent (abelian in the constant curvature case) and from Tits’ theorem [Ti] implying that every virtually nilpotent group satisfies (3).

Another important example one obtains by considering elementary subgroups (i.e. virtually cyclic) of word-hyperbolic (Gromov) groups which are also elementary according to the axioms (1-4).

We are going now to study hierarchies (i.e. strong accessibility) of finitely presented groups over elementary subgroups. A finite hierarchy of G over elementary subgroups is now defined as follows ([De-Po]):

Definition 3.2. Let G be a group and C a family of elementary subgroups of G. Let G = G^{0}. If G do not split as an amalgam or an HNN-extension over a subgroups in C, we stop. If not, we split G^{0} = G_{1}^{1} ∗_{C} G_{2}^{1}, or G = G_{1}^{1}∗_{C} (in this case, set G_{2}^{1} = G_{1}^{1}). Repeating the splitting procedure for G_{1}^{1} and G_{2}^{1} over elementary subgroups, we get groups G_{1}^{2}, G_{2}^{2}, G_{3}^{2}, G_{4}^{2}, and so on. A finite hierarchy for G is a splitting procedure which terminates, i.e. there is a step i such that none of the groups G_{j}^{i} splits as an amalgam or an HNN-extension over a group in C.

We will say that the group G has a finite hierarchy relatively to the sub-system E of C (in other words, (G, E) has a hierarchy) if G has a finite hierarchy over elementary subgroups such that all the subgroups in E are elliptic in every decomposition appearing in this hierarchy.

The length of the hierarchy is the number of decompositions needed to reach the final one (i.e. when all vertex groups are indecomposable over subgroups in C or are in C). We define then the number l(G) to be the minimal number of the lengths among all finite hierarchies of G. Similarly l(G, E) denotes the minimal number of the lengths of all hierarchies of G relatively to E.

The proof of the next Theorem closely follow the proof of the main Theorem 3.6 of the paper [De-Po] by keeping track of the peripheral system E:

Theorem C. Suppose G is a finitely presented group without 2-torsion and C ⊂ G is an elementary family satisfying the above axioms (1) − (4). Let E = {E_{1}, ...E_{k}} be a fixed finite subset of C then (G, E) has a finite hierarchy over elementary subgroups.

In other words, there exists a decomposition of (G, E) as an amalgam (or an HNN-extension)

G = A ∗_{C} B, (G = A∗_{C}),

such that

max{l(A, A ∩ E), l(B, B ∩ E)} < l(G, E).  (3)

Proof: Recall that in order to prove Theorem 3.2 in [De-Po] we used a version of an invariant c(·) (called complexity) of finitely presented groups appeared first in [De1]. This notion can be easily adapted to the “relative” case. Namely, consider a simplicial developable orbihedron Π of dimension 2 whose fundamental group is G (see [Ha]) such that the vertex stabilizers of Π are in C and every subgroup E_{i} fixes a vertex x_{i} ∈ Π (i = 1, ..., k). We define first c(Π, E) to be the pair (T (Π), b_{1}(Π)), where T (Π) is the number of 2-dimensional faces of Π and b_{1}(Π) is the first Betti number of the underlying topological space of Π. Then c(G, E) is defined to be the infimum (for the lexicographical order) over all such G-orbihedra Π.

If now T is a (G, E)-tree, the main result of [De-Po, Theorem 3.2] produces a simplicial tree T̂ and a resolution f : T̂ → T so that the invariant c(·) of the vertex stabilizers of T̂ strictly decreases. All we need now is to check that the groups E_{i} are still elliptic on the tree T̂. To see this consider the orbihedron universal cover P of such a complex Π. The Axioms of Definition 3.1 allow one to construct a G-equivariant map ρ : P → T ∪ ∂T (see [De-Po, 4.1]). Recall that the tree T̂ is constructed to be the dual tree to the lamination Λ ⊂ P whose leaves are pre-images under ρ of the midpoints of the edges of T. Let E_{i} ∈ E be an elementary subgroup which fixes a vertex x_{i} ∈ P. By assumption it also fixes a vertex v_{i} in the tree T. As the map f is equivariant every element g ∈ E_{i} stabilizes a component Ω_{i} of P \ Λ which contains x_{i}. Thus the group E_{i} is contained in the stabilizer Gv̂_{i} of the vertex v̂_{i} corresponding to the component Ω_{i} which is a vertex stabilizer of T̂. The result now follows by repeating the argument of [De-Po, Theorem 3.6]. ■

We will now establish the following finiteness Theorem for acylindrical splittings over elementary subgroups :

Acylindricity Theorem (acylindrical superaccessibility relative to a subset). Let G be a finitely presented, and E_{1}, ..., E_{q} a fixed finite set of infinite elementary subgroups of G. Suppose that the pair (G, E) is one-ended and there is a finite bound for orders of finite subgroups of G. Then for each K ∈ ℝ there exists a finite number of G-trees T_{1}, ..., T_{M} such that all subgroups E_{i} are elliptic on T_{j}, and for every minimal (K, Φ)-acylindrical G-tree T for which E_{i} are elliptic on T, there exist an automorphism φ of G sending each group E_{i} into itself and a resolution φ^{∗}(T_{i}) → T (i ∈ {1, ..., M}).

Remark. This Theorem in the torsion-free case (i.e. for K-acylindrical splittings) in the absolute form (i.e. without the claim about subgroups E_{i}) was proved by Z. Sela [Se1]. Then T. Delzant [De] showed it for (K, Φ)-splittings. Following the argument of the paper [De] we are going to extend the result and to show that the automorphism φ sends subgroups E_{i} into themselves provided that one considers (K, Φ)-acylindrical splittings of G in which the subgroups E_{i} are elliptic (i = 1, ..., q).

Proof: We will briefly sketch the construction of the paper [De] emphasizing the presence of the subgroups E_{i}. Let Π be finite two dimensional CW-complex with π_{1}(Π) ≅ G all of whose 2-faces are either digons or triangles. Suppose also that Π contains subcomplexes B_{i} (i = 1, ..., q) whose fundamental groups are isomorphic to E_{i}. Following the paper [De] we construct a G-equivariant simplicial map ρ : P → T where P is the universal cover of Π. Let Λ̃ denote a lamination of Π whose leaves are preimages under ρ of the middle points of the edges of T. By construction, Λ̃ is a G-equivariant lamination and let Λ denote Λ̃/G. We now define a subgraph Λ_{k} of Λ by describing its intersection with each face Δ of Π. Namely Δ ∩ Λ_{k} are those leaves of Λ in Δ whose image under ρ is situated within a distance at least k from the images of the vertices or the center of Δ. It is proven in [De] (see Lemma 1.5) that the fundamental group of each connected component of Λ_{k} fixes pointwise a segment of the length k when acting on T. It follows by assumption that the fundamental group of each connected component of Λ_{k} is finite.

We now collapse all leaves of Λ_{k} as well as all complexes B_{i} to points. As the number of faces of Π and leaves in Λ \ Λ_{k} is uniformly bounded one can show that the number of faces and edges of the resulting orbihedron Π′ is uniformly bounded too (here one uses essentially the minimality of the tree T see [De, Lemmas 2.1, 2.2]). Each vertex stabilizer of Π′ is either finite or is one of the groups E_{i}. As the orders of finite subgroups of G are uniformly bounded, there are only finitely many orbihedrons with all these properties, so Π′ must belong to a finite set of orbihedrons {Ω_{1}, ..., Ω_{M}}, with M depending only on the group G and the system of its subgroups E_{i} (i = 1, ..., q).

There exists a simplicial map θ_{k} between the complexes Π′ and Ω_{k} (for some k ∈ {1, ..., M}). This map induces the isomorphism (θ_{k})_{∗} : G → π_{1}^{o}(Ω_{k}) where π_{1}^{o}(Ω_{k}) is the orbihedron fundamental group of Ω_{k}. Notice that θ_{k} lifts to an equivariant map θ̃_{k} between P′ and Ω̃_{k} which are the orbihedron universal covers of Π′ and Ω_{k} correspondingly. If θ̃_{k}(x_{i}) = y_{j} where the stabilizer of the point x_{i} ∈ P′ is E_{i} and y_{j} ∈ Ω_{k} (i, j ∈ {1, ..., q}, k ∈ {1, ..., M}) then we have (θ_{k})_{∗}(C_{j}) ⊂ Stab(y_{j}) = C_{j}. Up to taking a power of θ_{k} and retaining the notation we may suppose that (θ_{k})_{∗}(E_{i}) ⊂ E_{i}. Following [De, Thm 3.1] denote by τ̂ the dual tree to the lamination which is the image of the lamination Λ in P′ and let T_{k} be the same for the orbihedron Ω_{k}. Similarly to the proof of Theorem C we obtain that the groups E_{i} are all elliptic on the tree τ̂ and there is an equivariant simplicial map τ̂ → T_{k}. The actions of the groups G and π_{1}^{o}(Ω_{k}) on the trees τ̂ and T_{k} respectively are conjugate by the map θ_{k}. Thus we find that τ̂ = θ_{k}^{∗}(τ_{k}) and the Theorem follows. ■

4 Splittings of Kleinian Groups

We start with the following important

Lemma 4.1. Let G be a finitely presented Kleinian group and E = {E_{1}, ..., E_{k}} be a family of elementary subgroups of G. Suppose that the pair (G, E) is one-ended. Then the pair (G, E) does not have essentially non-maximal splittings if and only if each irreducible (G, E)-splitting over elementary subgroups is (K, Φ)-acylindrical for some uniform constant K > 0 depending only on G. Furthermore, every essentially non-maximal splitting of (G, E) as an amalgam or an HNN-extension is (3, Φ)-acylindrical.

Proof: To prove the sufficiency we use essentially the same argument as in the papers [De] and [Se]. Suppose that the pair (G, E) has no essentially non-maximal splittings and let G act on a simplicial tree T with elementary edge stabilizers. Then G splits as the graph of groups X = T/G. Let m denote the number of edges of X. We will first show that the tree T is 2m+1-acylindrical. To this end suppose that l is an embedded path in T consisting of n successive edges such that n ≥ 2m+1. We want to show that the stabilizer C of l is a finite group. Arguing by contradiction suppose that the group C is infinite. As n ≥ 2m + 1 the path l contains at least three distinct edges e_{1}, e_{2}, e_{3} in the same G-orbit. Let C_{i} be the stabilizer of the edge e_{i} and let α_{i} and α′_{i} be its vertices (i = 1, 2, 3). Let e_{2} = g(e_{1}) and e_{2} = h(e_{3}) for some elements g and h not belonging

to C_{1} and C_{3} respectively. We have C ⊂ ⋂_{i=1}^{3} C_{i} and C_{2} = gC_{1}g^{−1}, C_{2} = hC_{3}h^{−1}.

As g^{−1}C_{2}g ∩ C_{2} ⊃ C and C is infinite we deduce that g^{−1}C̃_{2}g = C̃_{2} where C̃_{2} is the unique maximal elementary subgroup of G containing C_{2}; the same property holds for h. Then the elements g and h belong to C̃_{2} which also contains C. As G does not have essentially nonmaximal splittings, it follows that C̃_{2} fixes a point on the tree T and so there is a vertex v ∈ T whose stabilizer D contains C̃_{2}.

Let [α′_{i}, α_{i+1}] denote the segment of the path l between the vertices α′_{i} and α_{i+1}. A standard argument [Se, I-6.4] shows that either the element g fixes a point x in [α′_{1}, α_{2}] or g acts on T without fixed points. We have already shown that the latter case is impossible. Similarly, the element h fixes a point y ∈ [α′_{2}, α_{3}]. Now their common fixed point v belongs to the same connected component of T \ e_{2} as one of the vertices x or y, say x. Then the element g fixes pointwise the path in T between the vertices y and v which contains the edge e_{1}, and so g ∈ C_{1} which is impossible. Thus n ≤ 2m. In particular, if the graph X contains only one edge the splitting X (i.e. amalgam or an HNN-extension) is (3, Φ)-acylindrical.

By the result of M. Bestvina-M. Feighn [BeF2] there is a uniform upper bound ν(G) < ∞ for the number of edges of all irreducible splittings of G with elementary edge stabilizers. Thus, setting K = 2ν(G) + 1 we obtain the result. The necessary condition is proved.

Suppose now that the group has an essentially non-maximal splitting G = π_{1}(X, C_{e}, G_{v}) relatively to the system E. As the pair (G, E) is one ended, every edge group C_{e} = C of X is an infinite elementary subgroup of infinite index in C̃. Therefore C̃ is an infinite parabolic subgroup of G, and C̃ does not fix any point in T - universal cover of X. Since the group C̃ is virtually abelian it then follows from [S, 6.5, Proposition 27] that there is an element t in C̃ acting hyperbolically on T. The group C̃ contains an abelian subgroup of finite index C′ and, so there exists k ∈ ℕ that t^{k} ∈ C′, and t^{k} centralizes the group C_{0} = C ∩ C′. Therefore, the group generated by C_{0} and t^{k} fix also the edges e, t^{k}(e), ..., t^{nk}(e), ..., and, hence a segment of arbitrary big length. We get that the graph X is not (K, Φ)-acylindrical for any K ∈ ℕ. The Lemma follows. QED.

The Acylindricity Theorem proved in the previous Section will be used to obtain the following finiteness result about splittings of Kleinian groups:

Theorem D. Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary, geometrically finite, Kleinian group without 2-torsion and E = {E_{1}, ..., E_{k}} be a family of elementary subgroups of G. Suppose that the pair (G, E) is one-ended and does not have essentially non-maximal splittings over elementary subgroups. Then up to an automorphism of (G, E) there exist at most finitely many minimal splittings of G over elementary subgroups.

Proof: The argument is essentually due to Sela [Se]. Suppose that ρ : T̂ → T is a resolution, then according to [BeF1] there exists a sequence of Stallings folds:

T̂ = T_{0} → T_{1} → T_{2} → ... → T_{m}.

There are several types of folds, each of these operations either strictly decreases the number of edges of T_{i} (modulo G) or strictly increases the edge stabilizer of the edge along which the folding is done. Now for any edge e we have either rank(C_{ρ(e)}) > rank(C_{e}) or |C_{ρ(e)} : C_{e}| < ∞. In both cases the group C_{ρ(e)} contains C_{e} and is contained in the maximal elementary subgroup C̃_{e}. We note that if Ẽ ⊂ G is a maximal elementary subgroup containing E then there does not exist an infinite ascending sequence of distinct subgroups of Ẽ (one easily checks this first for a maximal abelian subgroup of finite index in C̃_{m+k} and then deduces it for C̃_{m+k}). Consequently one can not make infinitely many Stallings folds without changing the rank of the subgroup E = C_{e}. However rank E_{i} ≤ rank Ẽ < ∞. Thus, for a given edge e the number of Stallings folds is finite. As the number of edges of the tree T_{0} is finite modulo G-action, the number of G-trees obtained from T_{0} by doing Stallings folds is also finite.

Lemma 4.1 now yields that every G-splitting over elementary subgroups is (3, Φ)-acylindrical. It follows from the Selberg Lemma [Ra] that G ⊂ GL_{n}(ℂ) contains a torsion free subgroup of finite index ϕ; in particular every finite subgroup of G has order less than ϕ. Then by the Acylindricity Theorem every (K, Φ)-splitting of G over elementary subgroups can be obtained from one of the trees T_{i} by doing a finite number of Stallings folds and up to an automorphism of G preserving the groups C_{j}. Combining all these facts we conclude that there are at most finitely many irreducible splittings of G over elementary subgroups. Theorem D is proved. ■

One more accessibility result which we are now going to prove will deal with refined sequences of splittings defined as follows:

Definition 4.2. Let F be graph of groups decomposition of the pair (G, E). We say that the graph F_{1} refines F if it is obtained from F by replacing of a vertex v ∈ F^{0} by a non-trivial graph of groups decomposition F_{v} of the pair (G_{v}, E ∩ S), where S is the set of edge groups of F.

A sequence {F_{n}} of graphs of groups decompositions of (G, E) is called a refining sequence if for every n the graph F_{n+1} refines F_{n}. We say that refining sequence {F_{n}} stabilizes if the exists n_{0} such that F_{n} = F_{n_{0}} for all n > n_{0}.

Proposition 4.3. Suppose G is a finitely presented Kleinian group and E is a fixed sub-system of elementary subgroups of G. Suppose that the pair (G, E) is one-ended and has no essentially non-maximal splittings. Then every refining sequence of decompositions of (G, E) stabilizes.

Remark 4.4. We thank M. Bestvina who informed us about a result of this type. In the paper [Bo] a similar statement is proved (Theorem 6.1) by a different method.

Proof: Suppose {F_{n}} is a sequence of decompositions of the pair (G, E) so that the graph F_{n+1} is obtained from F_{n} by doing an elementary refinement; i.e. the label of some vertex v of F_{n} is replaced by an elementary splitting A ∗_{C} B or A∗_{C}, in which all the edge groups of the graph F_{n} are elliptic. We have natural embedding F_{m} ⊂ F_{n} if n > m. Collapsing of a vertex is the inverse operation to the refinement. A vertex v of valence two of a graph of groups of (G, E) is said to be trivial if its label coincide with the label of one of the edges incident to it.

Bestvina-Feighn’s Accessibility Theorem [BeF2] ensures that there exists m such that all vertices in F_{n} \ F_{m} are trivial for n > m. Indeed, if it is not so then collapsing all trivial vertices in the graph F_{n} we will get an irreducible graph of groups decomposition of (G, E) with elementary edge stabilizers having an unbounded number of vertices (when n → +∞) which is prohibited by [BeF2]. Therefore if the sequence {F_{n}} does not stabilize then for some vertex group A_{m} of F_{m} we will have an infinite chain of elementary refinements:

A_{m} = A_{m+1} ∗_{C_{m+1}} C_{m}, A_{m+1} = A_{m+2} ∗_{C_{m+2}} C_{m+1}, ... A_{m+k} = A_{m+k+1} ∗_{C_{m+k+1}} C_{m+k} ... , (4)

where C_{m+k} is an infinite elementary subgroup of G as G splits over it and is one ended. One has C_{m+k} ⫌ C_{m+k+1} (as the splittings (4) are all non-trivial). It also follows that for all but finitely many indices we have |C_{m+k} : C_{m+k+1}| < ∞ as the rank of the maximal elementary group C̃_{m+k} is finite. We deduce from (4):

A_{m} = C_{m} ∗_{C_{m+1}} C_{m+1} ∗_{C_{m+2}} C_{m+2} ∗ ... ∗_{C_{m+k}} C_{m+k} ∗ ... ,

and further

A_{m} = A_{m+1} ∗_{C_{m+1}} C_{m}; A_{m} = A_{m+2} ∗_{C_{m+2}} C_{m}; ... A_{m} = A_{m+k} ∗_{C_{m+k}} C_{m} ... .  (4′)

Let us first show that every decomposition of A_{m} in (4′) is essentially maximal. We shall check this for instance for the first one. Let T_{m+1} denote the G-tree - the universal cover of the graph F_{m+1}. Since G has no essentially non-maximal splittings, the maximal elementary subgroup C̃_{m+1} of G containing C_{m+1} fixes a point, say w ∈ T_{m+1}. Up to conjugation we may suppose that there is an edge e in T_{m+1} whose stabilizer is C_{m+1} and the vertices α and β of the edge e are stabilized by the groups A_{m+1} and C_{m} correspondingly. Denote by C′_{m+1} the maximal elementary subgroup of A_{m} containing C_{m+1} (C′_{m+1} = C̃_{m+1} ∩ C_{m+1}). As w belongs to a component of F_{m+1} \ e which contains α or β, say α, we obtain that C′_{m+1} fixes a path between w and β pointwise. This implies that C̃_{m+1} ⊂ A_{m+1} which shows that the splitting A_{m} = A_{m+1} ∗_{C_{m+1}} C_{m} is essentially maximal.

Then by Lemma 4.1 every elementary splitting in (4’) is (3, Φ)-acylindrical, moreover it is obviously minimal. The finiteness Theorem D for such splittings now tells that up to automorphisms of A_{m} there are only finitely many different splittings in (4′). Therefore, there exist indices k and l and an automorphism φ of A_{m} so that φ(A_{m+k}) = A_{m+l}, φ(C_{m+k}) = C_{m+l}, φ(C_{m+k−1}) = C_{m+l−1} (k < l). As the group C_{m+l} is infinite and C_{m+k} ⊃ C_{m+l}, we must have C̃_{m+k} = C̃_{m+l} for the maximal elementary subgroups. It follows that φ(C̃_{m+k}) = C̃_{m+l} = C̃_{m+k}, and whence C_{m+k} = C_{m+l}. Indeed, otherwise the virtually abelian group C̃_{m+k} would contain an infinitely increasing sequence of subgroups φ^{−n}(C_{m+l}) which is impossible. This is a contradiction. The Proposition is proved. ■

We call an edge in a graph of groups decomposition of G trivial if its label coincide with the labels of both its vertices.

Definition 4.5. Let G be a finitely presented group endowed with a system C of elementary subgroups. We define the number ν(G, E) to be the maximal number of non-trivial edges among all (G, E)-splittings with elementary edge stabilizers, where E ⊂ C is a fixed sub-system of elementary subgroups.

Corollary 4.6. Let G be a finitely presented group satisfying the assumptions of Proposition 4.3. Suppose it has a non-trivial (G, E)-splitting G = π_{1}(X, G_{v}, C_{e}) then for every vertex v ∈ X we have:

ν(G_{v}, E ∪ C) < ν(G, E).

Proof: Proposition 4.3 implies that ν(G_{v}, E ∪ C) < +∞. Thus we can find a (G_{v}, E ∪ C)-splitting F_{A} having the maximal number ν(G_{v}, E ∪ C) of non-trivial edges. We then refine the splitting X by F_{A}. As the splitting X has at least one non-trivial edge, this refined splitting is a (G, E)- splitting which contains one more edge. QED.

Let G be a non cohopfian Kleinian group, and f : G → G is an injective non surjective endomorphism, then the result of the paper [OP] implies that G admits a non-trivial action on a simplicial tree with elementary edge stabilizers. The following Proposition provides a relative version of this result:

Proposition 4.7. Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary, geometrically finite Kleinian group and E = {E_{1}, ..., E_{k}} is a finite family of elementary subgroups of G. Suppose that the pair (G, E) is non-cohopfian and let f : (G, E) → (G, E) be an injective endomorphism which is not surjective. Then G has a non-trivial splitting over elementary subgroups relatively to E.

Proof: We may assume (w.l.o.g.) that all the subgroups E_{i} are infinite maximal elementary subgroups of G. For any finite system S of generators of G one can consider the following displacement function:

d_{m}(f, S, G) = min_{x∈ℍ^{n}} max_{s∈S} d_{ℍ^{n}}(x, f^{m}(s)(x)),  (5)

where d_{ℍ^{n}}(·) denote the hyperbolic distance of ℍ^{n}.

It is proved in the paper [OP] that if the map f is not surjective then for any generating system S the function d_{m}(f, S, G) is not bounded (m ∈ ℕ). In this case by the theorem of Bestvina-Paulin [Be], [Pa], the group G acts stably and non-trivially on a real tree T_{ℝ} with elementary edge stabilizers. Furthermore, it is proven in [Be], [Pa] that

\overline{lim}_{m→∞} \frac{l(f^{m}(g))}{d_{m}(f, S, G)} = L_{ℝ}(g),  (6)

where l(g) = inf d_{ℍ^{n}}(x, g(x)) and L_{ℝ}(g) = inf d_{T_{ℝ}}(x, g(x)) are the translation lengths in the hyperbolic space ℍ^{n} and on the tree T_{ℝ} respectively.

By Rips’ theorem [BeF1] it follows that there is a non-trivial simplicial G-tree T_{∞} with elementary edge stabilizers, which will be further called the limit tree of G corresponding to the map f.

If E ∈ E is a subgroup such that f^{m}(E) is a parabolic subgroup (∀ m > m_{0}) then it follows from (6) that E fixes a point in the tree T_{ℝ} and then the relative version of Rips’ theorem [BF1, Theorem 9.6] implies that E fixes a point in the limit tree T_{∞}.

Arguing by contradiction suppose that there exists an element γ ∈ E_{i} which acts hyperbolically on T_{∞}. Thus for every m ∈ ℕ the group f^{m}(E_{i}) is an infinite virtually cyclic loxodromic group (in fact f^{m}(E_{i}) ⊂ E_{i} and E_{i} is itself an infinite virtually cyclic loxodromic subgroup of G). Let us choose a generating system S = {γ, γ_{1}, ..., γ_{k}, a_{1}, ..., a_{l}} of G where γ_{i} are generators of subgroups E_{j} and the elements a_{j} do not belong to E.

We first claim that there exists an element b ∈ G \ E such that all but finitely many elements f^{m}(b) are not parabolic. To see this, choose α to be an element of infinite order such that α ∉ E and set b = αγα^{−1}. Then up to replacing α by some power if needed, we may assume that the element b does not belong to E. Indeed, otherwise there exists k ∈ ℕ such that α^{k} normalizes E_{i} and so α^{k} ∈ E_{i} by the maximality of E_{i}, which is impossible by the choice of α. As γ acts hyperbolically on the tree T_{∞}, the element b (being a conjugate of γ) acts hyperbolically on T_{∞} too. So by the above argument all but finitely many elements b_{m} = f^{m}(b) are loxodromic and we have l(b_{m}) → ∞. Moreover, b_{m} ∉ E since f(E) ⊂ E. We add the element b to the generating system retaining for it the same notation S.

Up to passing to a subsequence, we may choose an element g ∈ S and a point x_{m} ∈ ℍ^{n} which realizes the mini-max in (5):

d_{m}(f, S, G) = d_{ℍ^{n}}(x_{m}, f^{m}(g)(x_{m})).

As the element γ ∈ E acts hyperbolically on T_{∞} the quantity

L_{ℝ}(γ) = lim_{m→∞} \frac{l(f^{m}(γ))}{d_{ℍ^{n}}(x_{m}, f^{m}(g)(x_{m}))},

is strictly positive by the relative version of Rips’ theorem, so we have 0 < L_{ℝ}(γ) ≤ 1. Let A_{γ} denote the invariant axis of γ. As f(E_{i}) ⊂ E_{i} we see that A_{γ} is the invariant axis for every group f^{m}(E_{i}) (m ∈ ℕ), which will be important further. We have the following:

Lemma 4.8. Suppose that b ∈ G\E such that l(b_{m}) → +∞ (m → +∞) then dist_{ℍ^{n}}(A_{γ}, b_{m}(A_{γ})) is unbounded, where dist_{ℍ^{n}}(·) denotes the hyperbolic distance between subsets of ℍ^{n}.

Proof of the Lemma: Suppose not, and dist_{ℍ^{n}}(A_{γ}, b_{m}(A_{γ})) ≤ K < +∞ for some constant K. Let w_{m} ∈ A_{γ} and y_{m} ∈ b_{m}(A_{γ}) so that d_{ℍ^{n}}(w_{m}, y_{m}) = dist_{ℍ^{n}}(A_{γ}, b_{m}(A_{γ})). Let w′_{m} denote the point b_{m}^{−1}(y_{m}) ∈ A_{γ}, so d_{ℍ^{n}}(w_{m}, b_{m}(w′_{m})) ≤ K. Since the subgroup E_{i} acts co-compactly on the axis Aγ there exist elements c_{m} and c′_{m} in E_{i} so that w_{m} = c_{m}(z_{m}), w′_{m} = c′_{m}(z′_{m}), where z_{m}, z′_{m} ∈ A_{γ} and d_{ℍ^{n}}(z_{m}, z′_{m}) ≤ K_{1} < +∞ for some K_{1}. We deduce that d_{ℍ^{n}}(z_{m}, c_{m}^{−1}b_{m}c′_{m}(z_{m})) ≤ K + K_{1} < +∞. As the group G is discrete and c_{m}^{−1}b_{m}c_{m} = c_{m}^{−1}f^{m}(b)c′_{m} ∈ G (m ∈ ℕ) it follows that ∃m_{0} ∀m > m_{0} : c_{m}^{−1}b_{m}c′_{m_{0}} = c_{m_{0}}^{−1}b_{m_{0}}c′_{m_{0}} and, hence l(b_{m}) = l(b_{m_{0}}) < +∞, ∀ m ≥ m_{0}. Thus we obtain a contradiction with the fact that l(b_{m}) → +∞. The Lemma is proved. ■

To finish the proof of the Proposition we are going to show that γ can not act hyperbolically on T_{∞} contradicting our assumption. For the element b = αγα^{−1} ∈ G \ E, which satisfies the conclusion of the Lemma and the sequence b_{m} = f^{m}(b) we have

0 < lim_{m→∞} \frac{d_{ℍ^{n}}(x_{m}, b_{m}(x_{m}))}{d_{m}(f, S, G)} ≤ 1,

where x_{m} is the point realizing the mini-max in (5). There are two cases according to whether or not the quantity D_{m} = dist_{ℍ^{n}}(x_{m}, A_{b_{m}}) remains bounded (where A_{b_{m}} is the axis of the loxodromic element b_{m}).

Case 1. D_{m} is unbounded.

Since b ∈ S, so d_{ℍ^{n}}(b_{m}(x_{m}), x_{m}) < d_{m}(f, S, G). We have up to passing to a subsequence:

\frac{l(b_{m})}{d_{m}(f, S, G)} ≤ e^{−D_{m}} \frac{d_{ℍ^{n}}(b_{m}(x_{m}), x_{m})}{d_{m}(f, S, G)} → 0,

implying that the element b = αγα^{−1} acts elliptically on T_{∞} and, so is γ. A contradiction.

Case 2. D_{m} is bounded.

Choose z_{m} ∈ A_{b_{m}} such that d_{ℍ^{n}}(x_{m}, z_{m}) = dist_{ℍ^{n}}(x_{m}, A_{b_{m}}) = D_{m} ≤ const < +∞. From Lemma 4.8 we obtain that the quantity dist_{ℍ^{n}}(A_{b_{m}}, A_{γ}) tends to infinity (up to a subsequence). Choosing a point w_{m} ∈ A_{γ} realizing the distance dist_{ℍ^{n}}(x_{m}, A_{γ}) we have that M_{m} = d_{ℍ^{n}}(w_{m}, x_{m}) → +∞. For the elements b_{m} = f^{m}(α)f^{m}(γ)f^{−m}(α) we have that l(b_{m}) = l(f^{m}(γ)) and thus:

\frac{l(b_{m})}{d_{m}(f, S, G)} = \frac{l(f^{m}(γ))}{d_{m}(f, S, G)} = \frac{d_{ℍ^{n}}(w_{m}, f^{m}(γ)(w_{m}))}{d_{m}(f, S, G)} ≤ e^{−M_{m}} \frac{d_{ℍ^{n}}(x_{m}, f^{m}(γ)(x_{m}))}{d_{m}(f, S, G)} ≤ e^{−M_{m}} → 0.

and as before we obtain that the element γ acts elliptically on the tree T_{∞} contradicting our assumption. The Proposition is proved. ■

5 Finding of a G-tree invariant under endomorphism

Suppose that the pair (G, E) is not cohopfian for a fixed finite family E = {E_{1}, ..., E_{k}} of elementary subgroups of G. Then Proposition 4.7 tells us that (G, E) splits as an amalgam (or an HNN-extension) over an elementary subgroup. In the following Proposition we obtain infinitely many such G-trees:

Proposition 5.1. Let G be a non-elementary, geometrically finite, Kleinian group without 2torsion endowed with the system E. Then the following assertions are true:

1) If f : (G, E) → (G, E) is a non-surjective monomorphism. then there exists a (G, E)-tree τ so that for every n ∈ ℕ the tree f^{n}^{∗}(τ) is a non-trivial (G, E)-tree.

2) If in addition, the pair (G, E) is one-ended and has no essentially non-maximal splittings then there exists a (G, E)-tree J so that for every n ∈ ℕ the tree f^{n}^{∗}(J) is a non-trivial, \underline{(K, Φ)-acylindrical} (G, E)-tree for some uniform constant K.

Proof: We will prove the first part of the Proposition by the induction on the length l(·, E) of the hierarchy of (G, E) introduced in Definition 3.2. By Theorem C we have that (G, E) splits as G = A ∗_{C} B or G = A ∗_{C} (C ∈ C) so that

max{l(A, A ∩ E), l(B, B ∩ E)} < l(G, E).

Let T denote the Bass-Serre tree corresponding to this splitting. If for all n ∈ ℕ the tree f^{n}^{∗}(T) is a non-trivial G-tree set T = J; if not there exists m ∈ ℕ such that up to conjugation f^{m}(G) is a subgroup of A or B, say A. As f is injective the subgroup A is a non-elementary group and we must have f^{m}(A) ⫋ A. Note that since the edge group C of the graph T/G is a quasi-convex subgroup of G then by the proof of [IKa, Lemma 3.5] it follows that A and B are quasi-convex subgroups of G. Thus [Sw] now yields that A and B are geometrically finite groups, in particular, they are finitely presented groups.

Note that by Proposition 4.7 we have l(G, E) ≥ 1. So let us first check the statement of the Proposition when l(G, E) = 1. Consequently l(A, A ∩ E) = 0 and immediately we get a contradiction. Indeed, the pair (A, E ∩ A) is not cohopfian so Proposition 4.7 applied to A yields that A splits non-trivially relatively to A ∩ E. This shows that in this case the trees f^{n}^{∗}(T) are non-trivial (G, E)-trees (∀ n ∈ ℕ).

Suppose now that l(G, E) > 1; as f is a non-surjective monomorphism of (A, A ∩ E) and l(A, E ∩ A) < l(G, E) we can apply the induction assumption to A. Thus, there is a non-trivial tree T_{A} so that f^{n}^{∗}(T_{A}) is a non-trivial (A, E ∩ A)-tree for all n ∈ ℕ. Consequently, the tree τ = f^{∗}(T_{A}) considered as a G-tree (as f(G) ⊂ A), gives rise to the sequence of G-trees f^{n}^{∗}(τ) (n ∈ ℕ), which are all non-trivial A-trees while restricted to A. Whence f^{n}^{∗}(τ) is a non-trivial G-tree (∀ n ∈ ℕ).

If E ∈ E then by the induction assumption, the group E ∩ A is conjugate into a vertex stabilizer of τ, say G_{v}. Thus, f^{−n}(E ∩ A) is contained in the vertex stabilizer f^{−n}(G_{v}) of the tree f^{n}^{∗}τ. Since f(E_{i}) ⊂ E_{i} we obtain E_{i} ⊂ f^{−n}(E_{i} ∩ A) ⊂ f^{−n}(G_{v}). We have shown that the system E is elliptic on the trees f^{n}^{∗}τ, and therefore f^{n}^{∗}τ is a non-trivial (G, E)-tree for every n ∈ ℕ. The first part is proved.

The graph τ/G may happen to be reducible. In this case we collapse in τ/G every edge whose label is equal to the label of one of its vertices. Denote J the universal cover (in the sense of Serre) of this new graph of groups decomposition of G. As G acts on τ without global fixed points, then obviously, G also acts without global fixed points on J, and the system E is elliptic on J. Furthermore, as the set of the non-elementary vertex stabilizers of the trees J and τ is the same, it follows from Part 1) that the trees f^{n}^{∗}J are also non-trivial (G, E)-trees (∀ n ∈ N). The pair (G, E) is one-ended and has no essentially non-maximal splittings, so Lemma 4.1 now yields that the reduced (G, E)-tree J is (K, Φ)-acylindrical for some uniform constant K. It is now easy to see that every tree f^{n}^{∗}J is (K, Φ)-acylindrical for the same constant K (n ∈ ℕ). Indeed, if not then there is a segment l of length K on the tree f^{n}^{∗}(J) whose pointwise stabilizer is an infinite subgroup C. Then f^{n}(C) is an infinite subgroup fixing pointwise the segment l on J too. This is impossible. The Proposition is proved.

■

The following Proposition is central in this Section:

Proposition 5.2. Let G ⊂ Isom_{+}ℍ^{n} be a discrete finitely presented group endowed with a fixed finite family E = {E_{1}, ...E_{k}} of elementary subgroups of G. Suppose that the pair (G, E) is oneended and does not have essentially non-maximal splittings. Let f : (G, E) → (G, E) be a nonsurjective monomorphism. Then there exist a non-trivial (G, E)-tree T with elementary edge stabilizers and a non-surjective monomorphism F : (G, E) → (G, E) such that F^{∗}T = T.

Proof: By the previous Proposition there exists a tree J so that f^{n}^{∗}J is a non-trivial (K, Φ)- acylindrical (G, E)-tree (∀n ∈ ℕ) for some uniform contsant K. A priori, the tree f^{n}^{∗}J is not a minimal G-tree. To be able to apply the finiteness Theorem D we choose a minimal G-subtree T_{n} of f^{n}^{∗}J. By construction of the latter it is straightforward that there exists a subtree T′_{n} of J invariant under the action of the group f^{n}(G) so that T_{n} = f^{n}^{∗}(T′_{n}). Furthermore T′_{n} ⊆ T′_{m} ⊆ J whenever n > m. It follows by Proposition 5.1 that T_{n} is a non-trivial G-tree which is still (K, Φ)-acylindrical for the same constant K. So Theorem D now yields that there exist integers n and m (n > m) and an automorphism φ of G so that:

φ^{∗}T_{n} = T_{m}, φ(E_{i}) ⊂ E_{i}.

This implies, in particular, that the G-equivalence classes of edges (resp. vertices) of the trees T_{n} and T_{m} are in 1 − to − 1 correspondence each other. Therefore T′_{n} = T′_{m} and we have φ^{∗}(f^{n}^{∗}(T′_{m})) = f^{m}^{∗}(T′_{m}). Obviously f^{n}^{∗}(T′_{m}) = f^{n−m}^{∗}(f^{m}^{∗}(T′_{m})), so we finally get

φ^{∗}(f^{n−m}^{∗}(f^{m}^{∗}(T′_{m}))) = f^{m}^{∗}(T′_{m}).

Setting T = T_{m} and F = f^{k} ∘ φ where k = n − m, we obtain

F^{∗}T = T

since (f^{n−m} ∘ φ)^{∗} = φ^{∗} ∘ (f^{n−m})^{∗}. The groups E_{i} are all elliptic on the trees (f^{m})^{∗}(J) (m ∈ ℕ) by Proposition 5.1 . As φ sends every subgroup E_{i} into itself the map F sends E_{i} into itself (i ∈ {1, ..., k}). Furthermore, F is injective and not surjective. The Proposition follows. ■

6 Proof of Theorem B.

Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary Kleinian group equipped with a finite system E = {E_{1}, ..., E_{k}} of elementary subgroups and suppose that F : (G, E) → (G, E) is a monomorphism of G sending each subgroup E_{i} into itself. Let Ẽ_{i} denote E_{i} if E_{i} is finite and the maximal elementary subgroup of G containing E_{i} if E_{i} is infinite.

The aim of this section is to prove :

Theorem B. Let G ⊂ Isom_{+}ℍ^{n} be a non-elementary, geometrically finite, Kleinian group without 2-torsion and E = {E_{1}, ..., E_{k}} be a family of elementary subgroups of G. Suppose that the pair (G, E) is one-ended. Then (G, E) is cohopfian if the following two conditions are satisfied:

1) The pair (G, E) does not have essentially non-maximal splittings over elementary subgroups.

2) The pair (G, E) does not split as an amalgam G = A∗_{C} C̃, so that C̃ is a maximal elementary subgroup of G and the normal subgroup of C̃ generated by C is of infinite index in C̃. ■

Proof: We will prove Theorem B by the induction on the invariant ν(·, E) which is maximal number of edges in all non-trivial (G, E)-splittings with elementary edge subgroups of G (see Definition 4.5). If ν(G, E) = 0 then it follows from Proposition 4.7 that every injective monomorphism of (G, E) is in fact surjective. So assume that the result is true for all groups having the complexity ν(·) less than that of G and satisfying the hypotheses of the Theorem.

Arguing by contradiction suppose that there exists a non-surjective monomorphism f : (G, E) → (G, E). Proposition 5.2 now yields that there exist a non-trivial (G, E)-tree T and a monomorphism F : (G, E) → (G, E) such that F^{∗}T = T. Furthermore the map F is surjective if and only if f is. Note that we may suppose (w.l.o.g.) that the graph Y = T/G is irreducible. Indeed, otherwise we collapse each edge whose label is equal to the label of the vertex incident to it. We obtain in this way an irreducible graph of groups which gives a non-trivial decomposition of G because the decomposition of G given by the graph Y was non-trivial. We retain the notations Y and T for this new graph of groups decomposition of G and its universal cover respectively.

The strategy of the further proof is the following: we first show that F is surjective on all non-elementary vertex stabilizers of T, then we prove the same for the elementary vertex stabilizers (using the condition 2) of the hypotheses. The surjectivity of F will then follow from the following Proposition:

Proposition 6.1. The map F : G → G is surjective if and only if it is surjective on each vertex group G_{v} of the graph Y.

Proof: If the graph Y is a tree of groups then the vertex groups G_{v} (v ∈ Y^{0}) generate the whole group G, and so the map F is surjective. Let us fix a maximal subtree of Y and prove the Proposition by induction on the number of edges which are not in the maximal subtree. If e is such an edge then G = A∗_{C_{e}} = ⟨A, C_{e} | tC_{e}t^{−1} = φ(C_{e})⟩, where φ : C_{e} → φ(C_{e}) is an isomorphism. The group A is the fundamental group of the graph Y′ of groups obtained from Y by deleting the edge e. Since Y′ has less number of non-separating edges than Y by the induction assumption, we have that F restricted to A is surjective. Denote H = C_{e} and H′ = φ(C_{e}). By Proposition 5.2 we have that F (H) ⊂ H and F (H_{1}) ⊂ H_{1}. Since tHt^{−1} = H_{1} we also obtain that F (t)F (H)F (t)^{−1} = F (H_{1}). Thus aHa^{−1} ∩ H ⊃ t^{−1}F (H_{1})t, where a = t^{−1} · F (t). As the group t^{−1}F (H_{1})t is infinite and is a subgroup of H, the element a belongs to the unique maximal elementary subgroup H̃ containing H. Then by our Definition 2.2 there exists a vertex v ∈ Y^{0} so that up to conjugation we have H̃ ⊂ G_{v} and, so a ∈ A. Since F is surjective on A there exists b ∈ A that F (b) = a. We finally obtain t = F (t · b^{−1}), and so F is surjective on G. The Proposition follows. ■

Denote by C the set of edge stabilizers of the tree T. For every C_{e} ∈ C we have F (C_{e}) ⊂ C_{e}. By Proposition 5.2 we also have that F (E_{i}) ⊂ E_{i}. As all edge groups of the graph Y = T/G are quasi-convex subgroups of G as we have already pointed out the results [IKa, Lemma 3.5] and [Sw] imply that every vertex stabilizer G_{v} is a geometrically finite group.

Every splitting of the pair (G_{v}, (C ∪ E) ∩ G_{v}) gives rise to an essentially maximal (G, E)- splitting. Thus by repeating the argument used in the proof of Proposition 4.3 we obtain that every (G_{v}, (C ∪ E) ∩ G_{v})-splitting is essentially maximal. Similarly, the pair (G_{v}, (C ∪ E) ∩ G_{v}) is one-ended. Since the graph Y defines a non-trivial splitting of G, Corollary 4.6 now implies that ν(G_{v}, (C ∪ E) ∩ G_{v}) < ν(G, E). As F |_{G_{v}} sends all subgroups in (C ∪ E) ∩ G_{v} into themselves the induction hypothesis yields that F |_{G_{v}} is surjective for every non-elementary vertex stabilizer G_{v} of the tree T.

Let us now consider edge groups and elementary vertex groups of the graph Y = T/G. Since our group G is non-elementary, collapsing every two neighboring vertices v_{1} and v_{2} whose vertex groups are elementary we still get a non-trivial splitting of G satisfying all the above properties. Similarly if there is a vertex v whose vertex group G_{v} is elementary and such that there is a loop e emanating from v, then we collapse this loop to v, the resulting vertex group will be still elementary and the map F sends it into itself. So we may suppose that every edge e ∈ Y^{1} which is not a loop has at least one vertex v ∈ ∂e whose label is a non-elementary group G_{v}. Also there is no loop of Y emanating from a vertex whose label is elementary.

We can also suppose that our graph Y does not have vertex groups which are non-maximal elementary groups. Indeed, if G_{v} is such a group then since G does not have essentially nonmaximal splittings the maximal elementary subgroup G̃_{v} containing G_{v} is contained in some other vertex group, say G_{v′}, however then the group G_{v} is contained in the stabilizer of the edge belonging to the path L between v′ and v. This contradicts the irreducibility of the graph Y.

Let us now prove that the restriction F |_{C_{e}} on every edge C_{e} is surjective. Indeed, as F is injective the group F^{−1}(C_{e}) is elementary (being isomorphic to F (F^{−1}(C_{e})) which is a subgroup of C_{e}) and we have F^{−1}(C_{e}) ⊃ C_{e} since F (C_{e}) ⊂ C_{e}. Let v_{1} ∈ ∂e be one of the vertices of e whose stabilizer G_{v_{1}} is not elementary. Since F |_{G_{v_{1}}} is surjective, for any y ∈ C_{e} there exists x ∈ G_{v_{1}} so that F (x) = y. If another vertex v_{2} ∈ ∂e has also a non-elementary stabilizer then by the same reason and the injectivity of F we obtain that x ∈ G_{v_{2}} and so x ∈ C_{e}. If now G_{v_{2}} is a maximal elementary subgroup of G we have F^{−1}(G_{v_{2}}) ⊃ F^{−1}(C_{e}) ⊃ C_{e} and then F^{−1}(G_{v_{2}}) = G_{v_{2}} since C_{e} is infinite and is contained in G_{v_{2}}. This shows that x ∈ G_{v_{2}} and again x ∈ C_{e}.

Let us now prove that F is surjective on every elementary maximal vertex group E_{v} (v ∈ Y^{0}) of the graph Y. Let e_{i} (i = 1, ..., l) be the edges incident to v and C_{i} be their labels. As there is no loop emanating from the vertex v it is now easy to see that we have a decomposition G = A∗_{C_{v}} E_{v} where C_{v} is the elementary group generated by C_{i} (i = 1, ..., l) and A is the fundamental group of the graph Y \ {v}. We already know that our map F is surjective on the edge group C_{i} and so it is surjective on C_{v}. Moreover applying the previous argument to every edge stabilizer of the tree T, we conclude that the restriction of F on every G-conjugate of C_{e} is also surjective. It now follows that F is surjective on the smallest normal subgroup N_{v} generated by C_{v}. Since F maps the group E_{v} into itself, and N_{v} onto itself, it induces an injective map Φ : E_{v}/N_{v} → E_{v}/N_{v}. By hypothesis the group E_{v}/N_{v} is finite, whence the map Φ is surjective, and so F is surjective on E_{v}. We have proved that F is surjective on every edge and vertex stabilizer of the graph Y.

From Proposition 6.1 we now derive that the map F : G → G is an automorphism of G. Then our initial map f : G → G is an automorphism too. To finish the proof we only need to show that f|_{E} is surjective on every E ∈ E. As f(E) ⊂ E the conclusion is obvious when E is finite. If it is not the case then by the uniqueness of the the maximal elementary subgroup Ẽ of G containing E we have f(Ẽ) = Ẽ as f(E) is an infinite subgroup of both. Again using the fact that an increasing sequence of subgroups of a virtually abelian group of a finite rank must stabilize we deduce that f(E) = E. Theorem is proved. ■

7 Necessary Condition in Theorem A.

The aim of this section is to prove the necessary condition in Theorem A which follows directly from the following result:

Theorem E. A finitely generated discrete group G ⊂ Isom_{+}ℍ^{n} is not cohopfian if one of the two conditions below is satisfied:

1) G has an essentially non-maximal splitting

G = π_{1}(X, G_{v}, C_{e}), where each vertex group C_{e} is elementary  (1′)

2) The group G splits as an amalgam G = A∗_{C} C̃, so that C̃ is a maximal elementary subgroup of G and the normal subgroup of C̃ generated by C is of infinite index in C̃.

We start with:

Remarks 7.1. 1) Examples of discrete geometrically finite groups in ℍ^{n} which are described in 1) and 2) of Theorem E do exist and are given in [OP].

2)If every elementary group over which G splits is in fact abelian and if there exists a splitting X described in Condition 2) then there is another splitting over elementary subgroups which is essentially non-maximal : the maximal elementary vertex group can be written as a central HNNextension with a base containing all corresponding edge stabilizers. However, this is not the case in general, as there exists (torsion-free) virtually abelian groups with finite abelianization.

We will first study essentially non-maximal splittings of G:

Proposition 7.2. Suppose G splits as a graph of groups (1’), where one of the edge groups C_{e} = E of the splitting (1’) is essentially non-maximal, and let Ẽ be the maximal elementary subgroup of G containing E with infinite index so that Ẽ is hyperbolic in the splitting (1’). Then there exists an element g ∈ Ẽ so that g centralizes E and g^{n} ∉ E (n ∈ ℕ).

Proof of 7.2: Let T be the Bass-Serre tree corresponding to the splitting (1’). The group Ẽ contains a normal abelian subgroup Ã of finite index. As Ẽ acts on T hyperbolically it follows that the group Ã also does. Hence by [S, I-6.5, Proposition 27] it follows that Ã leaves a line L ⊂ T invariant. As Ã is normal in Ẽ the group Ẽ also leaves L invariant. Then either Ẽ acts by translations on L; or it acts dihedrally on L (permuting the end points of L). So, there is a projection η of Ẽ on ℤ or onto ℤ_{2} ∗ ℤ_{2}. Moreover since the subgroup E of Ẽ fixes an edge e in T it fixes the axis L pointwise. So we may suppose that e ⊂ L and that E is the kernel of η (which is the kernel of the action of Ẽ on L). It follows that up to passing to a subgroup of index 2 and retaining the notation Ẽ for it, we have the following exact sequence:

0 ⟶ E ⟶ Ẽ ⟶^{η} ℤ ⟶ 1,

Let t denote the element of Ẽ which is mapped on the generator of ℤ. There exits m ∈ ℕ so that t^{m} ∈ A and up to replacing t by t^{m} and passing to a further subgroup of finite index we may suppose that t ∈ A. Also t^{n} ∉ E (∀n ∈ ℕ).

Let A denote the group Ã ∩ E which is a normal abelian subgroup of E of finite index. We have:

0 ⟶ A ⟶ E ⟶^{ξ} F ⟶ 1,  (∗),

where F is a finite group.

Definition. An automorphism of E will be called the automorphism of the sequence (*) if its restriction on A is trivial and if it induces the identity on F. The group of the automorphisms of (*) is denoted Aut(∗).

Let s : F → E be a set theoretic cross-section of ξ and ψ ∈ Aut(∗). Put c_{ψ}(f) = ψ(s(f)) ∘ s(f)^{−1}, ∀f ∈ F.

Lemma 7.3. The following assertions hold:

a) c_{ψ}(f) is a 1-cocycle of F taking values in A.

b) The map ψ → c_{ψ} is a group homomorphism.

Proof: a) Notice first that c_{ψ}(f) ∈ A since ψ induces the identity map on F. We have then: s(f · g) = s(f) · s(g) · α(f, g), where α(f, g) ∈ A.

ψ(s(f ·g))·(s(f ·g))^{−1} = ψ(s(f)s(g)α(f, g))·α^{−1}(f, g)(s(f)s(g))^{−1} = ψ(s(f))ψ(s(g))s(g)^{−1}s(f)^{−1}, since ψ(a) = a (∀a ∈ A). Further we derive : ψ(s(f ·g))·(s(f ·g))^{−1} = ψ(s(f))s(f)^{−1}+s(f)ψ(s(g))s(g)^{−1}s(f)^{−1} = ψ(s(f))s(f^{−1})+ρ(f)c_{ψ}(f) = c_{ψ}(f) + ρ(f)c_{ψ}(f), where ρ(f) denotes the action of f ∈ F on A given by conjugation by s(f). This proves a) by the definition of a cocycle (see [Br, p. 88]).

b) c_{ψ_{1}ψ_{2}}(f) = ψ_{1}ψ_{2}(s(f))s(f)^{−1} = ψ_{1}[ψ_{2}(s(f))s(f)^{−1}s(f)]s(f)^{−1} = ψ_{1}(c_{ψ_{2}}(f) · s(f))s(f)^{−1} = c_{ψ_{2}}(f) + c_{ψ_{1}}(f), here we used that c_{ψ_{2}}(f) ∈ A and that ψ_{1} keeps it unchanged. We have proved b). The Lemma is proved. ■

Proof of Proposition. Recall that t^{n} ∈ Ã \ E (∀ n ∈ ℕ). Let ψ be an interior automorphism of Ẽ given by the conjugation via t. As t acts identically by conjugation on A it is easy to verify that it also induces the identity on F, i.e.:

t̂ft̂^{−1} = f, ∀ f ∈ F,

where t̂= ξ(t). So ψ is an automorphism of the sequence (∗) and we get c_{ψ}(f) = ts(f)t^{−1}s(f)^{−1}. Since the group F is finite the first cohomology group H^{1}(F, A) is finite too and, so there exists p ∈ ℕ such that c_{ψ^{p}}(f) is a coboundary. It follows that there exists a ∈ A that c_{ψ^{p}}(f) = a − ρ(s(f)) · a = a + ρ(s(f))(−a). Writing this in the multiplicative form we have c_{ψ^{p}}(f) = as(f)a^{−1}s(f)^{−1} = t^{p}s(f)t^{−p}s(f)^{−1} implying that ∀ s(f) ∈ E : a^{−1}t^{p}s(f)t^{−p}a = s(f). Putting g = a^{−1}t^{p} we obtain that g is not trivial (t^{p} ∉ A) and centralizes E. The Proposition follows.

■

Note that above we also obtained the following fact which will be used further:

Remark 7.4. The group of the automorphisms of the sequence (*) is finite modulo conjugation in E (in other words the subgroup of Out(E) which preserves (*) is finite) ■.

Lemma 7.5. Suppose that the group G splits as a graph of groups G = π_{1}(X, G_{v}, C_{e}) with elementary edge stabilizers such that one of the edge group E = C_{e} is essentially non-maximal then G splits as an amalgam or an HNN-extension:

G = A ∗_{K} B, or G = A∗_{K},  (6)

where K is essentially non-maximal and contains E.

Proof:

Let T denote the tree which is the universal covering of X and let Ẽ be the maximal elementary subgroup of G containing E. The group Ẽ acting on T without fixed points has an invariant line L ⊂ T (see the beginning of the proof of Proposition 7.2). Since the subgroup E fixes a point in T it also fixes L pointwise. Let ê ⊂ L be an edge of T and α̂ and β̂ its vertices. We first claim that the stabilizer F of the edge ê in G coincide with the stabilizer K of ê in Ẽ (i.e. the kernel of the action of Ẽ on L). Indeed, both groups contain the group E which is an infinite group so by the uniqueness of the maximal elementary subgroup Ẽ containing E it follows that F ⊂ Ẽ which implies that F = K. Similar argument shows that the subgroup N of G leaving the line L invariant coincide with Ẽ. Indeed, the group N is elementary which follows from the fact that it has a projection to ℤ or ℤ_{2} ∗ ℤ_{2} whose kernel is the elementary group K, consequently N is an elementary group containing Ẽ, and thus Ẽ = N.

Let α, β, e denote the images in X of α̂, β̂, ê respectively under the projection p : T → X . First consider the graph X. Then the group G is the HNN-extension

G = A∗_{K} = ⟨A, t | tKt^{−1} = ϕ(K)⟩ where A is the fundamental group of the graph of groups Y = X \ e. Denote Ŷ the component of the preimage p^{−1}(Y) adjacent to the edge ê at the point α̂ ∈ T. Clearly p(Ŷ) = Y, so we may assume up to conjugation in G that the stabilizer of Y is A. As none G-translate of ê is contained in Ŷ and T is a tree we have L ∩ Ŷ = {α̂}. Let h ∈ Ẽ \ E be an element acting on L by translations. As h(α̂) ∈ L \ {α̂}, so h can not belong to the stabilizer of Ŷ. Consequently, the group Ẽ is hyperbolic in the splitting G = A∗_{K}.

The case when the edge e separates X is similar: we obtain the splitting G = A∗_{K} B where A and B are the fundamental groups of the graphs which are respectively connected components U and V of X \e. Denote Û and V̂ the components of p^{−1}(U) and p^{−1}(V) which are adjacent along the edge ê in T, in particular α̂ ∈ Û and β̂ ∈ V̂. The stabilizers Û and V̂ are up to conjugation the groups A and B. Again we have Û ∩ L = {α} and V̂ ∩ L = {β}. There exists an element h ∈ Ẽ \ K which acts by translations on L so h(α̂) ∈ L \ {α̂}, and the same for β̂. Consequently, the element h does not belong to the stabilizers of Û and V̂. This shows that Ẽ is not elliptic with respect to the splitting G = A ∗_{K} B. The Lemma is proved. ■

Proof of Theorem E : Let us first consider Condition 1) of the Theorem which is:

1) G has an essentially non-maximal splittings.

Then by Lemma 7.5 it follows that there is a splitting (6) of G as an amalgam or an HNNextension which is essentially non-maximal. The edge group K is an elementary essentially non-maximal subgroup and let K̃ be the maximal elementary subgroup of G containing K which is hyperbolic in the splitting (6). By Proposition 7.2 it follows that there exists an element t ∈ K̃ \ K which centralizes K.

Consider first the case of amalgamated product, i.e. G = A ∗_{K} B. Let us define the map f : G → G so that f(a) = tat^{−1}, ∀a ∈ A and f(b) = b. As t commutes with all elements from K the map f is obviously a homomorphism. If f(g) = 1 for some g ∈ G , then writing g in the normal form it is easy to see that g ∈ A or g ∈ B, so by injectivity of f on A and B we obtain g = 1. We have that f : G → G_{1} is an isomorphism where G_{1} = ⟨tAt^{−1}, B⟩ ≅ G. We need only to show that G_{1} ⫋ G. The group G_{1} being a subgroup of G acts on the Bass-Serre G-tree T corresponding to the splitting G = A ∗_{K} B. Denote by α and β the vertices of T whose stabilizers are A and B. Set d = min_{g∈G} dist_{T} (gtg^{−1}(α), α) then d > 1 since t is not conjugate into

A and B. In particular, t cannot normalize A (since otherwise tAt^{−1} fixes the path between t(α) and α pointwise and so tAt^{−1} is conjugate into K which is impossible).

The group G_{1} is the amalgamated free product tAt^{−1} ∗_{K} B and so every element g ∈ G_{1} can be written in the following form:

g = ta_{1}t^{−1} · b_{1} · ... · ta_{k}t^{−1} · b_{k}, or g = b_{1} · ta_{1}t^{−1} · ... · b_{k} · ta_{k}t^{−1}, a_{i} ∈ A \ K, b_{j} ∈ B \ K. (7)

As t centralizes the group K, so ta_{i}t^{−1} ∈ tAt^{−1} \ K and the formula (7) gives also a normal form of g in the amalgamated product G_{1} = tAt^{−1} ∗_{K} B. An easy calculation shows that

min_{α} dist_{T} (g(α), α) = 2k(d + 1), where α ∈ {a, b}.

It follows that the set L(G_{1}) of translation lengths of elements of G_{1} acting on T is equal to {2k(d + 1) | k ∈ ℕ} which is a proper subset of L(G) (e.g. 2 ∈ (L(G) \ L(G_{1})). This shows that G_{1} ⫋ G.

Consider now the case of an HNN-extension G = A∗_{K} = ⟨A, h | hKh^{−1} = ϕ(K)⟩ and let T be the corresponding Bass-Serre tree. There are two more subcases: a) h ∈ K̃ and b) h ∉ K̃. In the subcase a) we proceed as follows. By the proof of Proposition 7.2 there exist p ∈ ℕ and a ∈ A(K) so that the element g = h^{p} · a commutes with every element of K, where A(K) is the maximal abelian subgroup of K. Now we define the map f : G → G to be the identity on A and set f(h) = h^{p+1} · a. It is easy to check that f is an injective monomorphism (since h^{p+1} acts on K in the same way as h does) which is not surjective.

In the subcase b) we proceed similarly to the case of an amalgamated free product, namely put f(a) = tat^{−1}, ∀a ∈ A and f(h) = h, where t is an element in K̃ \ K acting hyperbolically on the tree T and centralizing K. Then any element g of the group G_{1} = f(G) = tAt^{−1}∗_{K} = ⟨tAt^{−1}, h|hKh^{−1} = ϕ(K)⟩ can be written as:

g = ta_{1}t^{−1} · h^{ε_{1}} · ta_{2}t^{−1} · ... · ta_{k}t^{−1} · h^{ε_{k}}, or g = h^{ε_{1}} · ta_{1}t^{−1} · h^{ε_{2}}ta_{2}t^{−1} · ... · ta_{k}t^{−1}, (7)

where ε_{i} ∈ ℤ and if ε_{i} < 0 and a_{i} ∈ K then ε_{i+1} ≤ 0, and if ε_{i} > 0 and a_{i} ∈ K then ε_{i+1} ≥ 0.

Let α be the vertex of T whose label is A and M be the line in T which is formed by the vertices h^{n}(α) (n ∈ ℤ). As h ∉ K̃ the element t does not belong to the maximal elementary subgroup containing h and so t(α) ∉ M. It is now straightforward that the translation length

l(g) of the element g in (7) is equal to 2kd + ∑_{i=1}^{k} |ε_{i}|, where d = min_{g∈G} dist_{T} (gtg^{−1}(α), α).

Indeed each term ta_{i}t^{−1} in (7) gives the contribution 2d and the term h^{ε_{i}} contributes |ε_{i}| to the

expression l(g). For a fixed d ≥ 1 the set {2kd + ∑_{i=1}^{k} |ε_{i}| | k ∈ ℕ, ε_{k} ∈ ℤ} is a proper subset of

ℕ which is the set of translation lengths of G acting on T. This shows that in this case G_{1} is a proper subgroup of G. Part 1) of Theorem E is proved.

Consider now Condition 2) of Theorem E which is:

The pair (G, E) splits as an amalgam G = A ∗_{C} C̃, so that C̃ is a maximal elementary subgroup of G and the normal subgroup of C̃ generated by C is of infinite index in C̃.

Suppose that X is such a graph of groups decomposition of G. We are going to construct a proper monomorphism from G into G which is the identity on all edge and vertex groups of X besides E_{v} and which sends E_{v} into itself being not surjective on it.

The group E_{v} is virtually abelian of finite rank, let A be a finite index normal free abelian subgroup of E_{v}. Denote C = A ∩ N_{v} and F = E_{v}/A. As N_{v} and A are normal in E_{v} the group C is normal in E_{v} too. Let also s : F → E_{v} be a normalized cross-section of the projection of E_{v} onto F. The group F acts on A by conjugation a → s(f)as^{−1}(f). Consider the vector space A ⊗ ℚ which we equip with a scalar product invariant under the induced action of F. As the subspace C ⊗ ℚ is invariant under the induced action of F there exists a subspace V in A ⊗ ℚ complementary to C ⊗ ℚ which is also invariant under this action. We can now find a subgroup B of A so that V = B ⊗ ℚ and so A ⊗ ℚ = (C ⊗ ℚ) ⊕ (B ⊗ ℚ). The group B has the following properties: B ∩ C = {id} (since A is torsion free); B is normal in E_{v}; and the group A′ = C ⊕ B is a normal free abelian subgroup of E_{v} of finite index. Denote F′ = E_{v}/A′. Consider the map h_{n} : A′ → A′ defined as h_{n}(c + b) = c + nb for every c ∈ C and b ∈ B. One can now find a group H_{n} and a homomorphism φ_{n} : E_{v} → H_{n} so that the following diagram commutes :

0 ⟶ A′ ⟶^{i} E_{v} ⟶^{p} F′ ⟶ 1
h_{n}↓  ^{φ_{n}}↓  ∥
0 ⟶ A′ ⟶^{i_{n}} H_{n} ⟶^{p_{n}} F′ ⟶ 1.

In fact the group H_{n} is the largest quotient of A′ ⋊ E_{v} such that the left-hand square of the above diagram commutes (see e.g. [Br, page 94]). Note that φ_{n} is injective and not surjective since h_{n} is ( ∀n ∈ ℕ). Furthermore, φ_{n} is the identity on C. We need now to show that H_{n} is isomorphic to E_{v}. Let us define a set-theoretic cross section s_{n} : F′ → H_{n} of the projection p_{n} (see the diagram) to be s_{n} = φ_{n} ∘ s. It is known (see e.g. [Br, III.3.12]) that the equivalence classes of extensions of A′ by F′ are in 1 − to − 1 correspondence with the elements of H^{2}(F′, A′). If α ∈ H^{2}(F′, A′) is the element which corresponds to the upper row of the commutative diagram then it satisfies i(α(g, γ))s(gγ) = s(g)s(γ) (g, γ ∈ F′). We have

s_{n}(g)s_{n}(γ) = φ_{n}(s(g)φ_{n}(s(γ)) = φ_{n}(s(g)s(γ)) = φ_{n}(i(α(g, γ))s(gγ)) = i_{n}(h_{n}(α(g, γ)))s_{n}(g, γ).

Setting α_{n}(g, γ) = h_{n}(α(g, γ)) we obtain from the above identity that α_{n}(g, γ) is an element of H^{2}(F′, A′). Since α(g, γ) takes its values in A′ we can write α(g, γ) = c(g, γ) + b(g, γ) where c(g, γ) ∈ C and b(g, γ) ∈ B. We also have the following commutative diagram:

0 ⟶  A′  ⟶^{i} E_{v} ⟶ F′ ⟶ 1
π_{1}↓  ^{π_{2}}↓  ∥
0 ⟶ B = A′/C ⟶^{j} E_{v}/C ⟶ F′ ⟶ 1

where π_{i} are the natural projections and j is the natural inclusion. Define a section σ : F′ → E_{v}/C to be σ = π_{2} ∘ s. Similarly we then show that σ(g)σ(γ) = j(p_{2}(α(g, γ)))σ(g, γ) (g, γ ∈ F′). As b = π_{1} ∘ α we obtain that b is also an element of H^{2}(F′, A′). Since the group F′ is finite it follows from [Br, III.10.2] that the group H^{2}(F′, A′) is annihilated by |F′|. Choosing n to be n ≡ 1(mod |G|) we obtain that b = nb and so α_{n} = c+b_{n} = c+b = α. This implies that α and α_{n} define the equivalent extensions and, so the groups E_{v} and H_{n} are isomorphic. Let n_{0} = |F′| + 1 then the map φ_{n_{0}} is an endomorphism of E_{v} which is injective, non-surjective and is the identity on C. Since C is an abelian normal subgroup of N_{v} it follows that φ_{n_{0}} is the identity on N_{v} too (because if h represents a coset in N_{v}/C then the element φ_{n_{0}}(h) · h^{−1} commutes with all elements in C and so φ_{n_{0}}(h) ≡ h(mod C)). By Remark 7.4 it follows that

∃ k ∈ ℕ, ∃ a ∈ C ∀h ∈ N_{v} : φ_{n_{0}}^{k}(h) = aha^{−1}.

Setting F |_{E_{v}} = a^{−1} · φ_{n_{0}}^{k} · a, we obtain an injective, not surjective endomorphism of E_{v} which is the identity on N_{v}. Extending now F by the identity to the fundamental group of the graph X \ {v} we obtain a monomorphism F : G → G which is injective and not surjective. Theorem E is proved. ■

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Thomas Delzant IRMA, Université L. Pasteur, 7 rue R. Descartes, F-67084 Strasbourg Cedex, France, delzant@math.u-strasbg.fr Leonid Potyagailo: UFR de Mathématiques, Université de Lille 1, 59655 Villeneuve d’Ascq cedex, France; potyag@gat.univ-lille1.fr