Acylindrical Accessibility for Groups Acting on R-Trees

ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES

ILYA KAPOVICH AND RICHARD WEIDMANN

Abstract. We prove an acylindrical accessibility theorem for ﬁnitely generated groups acting on R-trees. Namely, we show that if G is a freely indecomposable non-cyclic k-generated group acting minimally and M- acylindrically on an R-tree X then there is a ﬁnite subtree Y ⊆ X of measure at most 2M(k − 1) + such that GY = X. This generalizes theorems of Z.Sela and T.Delzant about actions on simplicial trees.

1. Introduction An isometric action of a group G on an R-tree X is said to be M- acylindrical (where M ≥ 0) if for any g ∈ G, g 6= 1 we have diam Fix(g) ≤ M, that is any segment fixed point-wise by g has length at most M. For example the action of an amalgamated free product G = A ∗C B on the corresponding Bass-Serre tree is 2-acylindrical if C is malnormal in A and 1-acylindrical if C is malnormal in both A and B. In fact the notion of acylindricity seems to have first appeared in this context in the work of Karras and Solitar [32], who termed it being r-malnormal. Sela [37] proved an important acylindrical accessibility result for finitely generated groups which, when applied to one-ended groups, can be restated as follows: for any one-ended finitely generated group G and any M ≥ 0 there is a constant c(G, M) > 0 such that for any minimal M-acylindrical action of G on a simplicial tree X the quotient graph X/G has at most c(G, M) edges. This fact plays an important role in Sela’s theory of JSJ- decomposition for word-hyperbolic groups [38] and thus in his solution of the isomorphism problem for torsion-free hyperbolic groups [36]. More- over, acylindrical splittings feature prominently in relation to the Combi- nation Theorem of Bestvina-Feighn [7, 9] and its various applications and generalizations [17, 23, 26, 33]. Unlike other kinds of accessibility results, such as Dunwoody accessibility [20, 21], Bestvina-Feighn generalized accessibility [5, 6] and the strong accessibility (introduced by Bowditch [13] and proved by Delzant and Potyagailo [19]), the acylindrical accessibility holds for finitely generated and not just finitely presented groups.

Date: October 16, 2002. 2000 Mathematics Subject Classiﬁcation. 20F67. The ﬁrst author was supported by the U.S.-Israel Binational Science Foundation grant BSF-1999298. 1 2 I. KAPOVICH AND R.WEIDMANN

Delzant [18] obtained a relative version of Sela’s theorem for finitely presented groups with respect to a family of subgroups. In particular, he showed that the constant c(G, M) above can be chosen to be 12MT , where T is the number of relations in any finite presentation of G where all relators have length three. Weidmann [40] used the theory of Nielsen methods for groups acting on simplicial trees to show that for any k-generated one-ended group one can choose c(G, M) = 2M(k − 1). In the present paper we obtain an analogue of this last result for groups acting on R-trees. Before formulating our main result let us recall the notion of Nielsen equivalence: Definition 1.1 (Nielsen equivalence). Let G be a group and let M = n (g1, . . . , gn) ∈ G be an n-tuple of elements of G. The following moves are called elementary Nielsen moves on M: −1 (N1) For some i, 1 ≤ i ≤ n replace gi by gi in M. (N2) For some i 6= j, 1 ≤ i, j ≤ n replace gi by gigj in M. (N3) For some i 6= j, 1 ≤ i, j ≤ n interchange gi and gj in M. n 0 n We say that M = (g1, . . . , gn) ∈ G and M = (f1, . . . , fn) ∈ G are 0 Nielsen-equivalent, denoted M ∼N M , if there is a chain of elementary Nielsen moves which transforms M to M 0. 0 0 It is easy to see that if M ∼N M then M and M generate the same subgroup of G. For this reason Nielsen equivalence is a very useful tool for studying the subgroup structure of various groups. We obtain the following statement which can be regarded as an “acylindrical accessibility” result for finitely generated groups acting on real trees. Indeed, our theorem says that there is a bound on the size of a “fundamen- tal domain” for a minimal M-acylindrical isometric action of a k-generated group on an R-tree: Theorem 1.2. Let G be a freely indecomposable finitely generated group acting by isometries on an R-tree X. Let M ≥ 0. Suppose that G 6= 1 is not infinite cyclic, that the action of G is M-acylindrical, nontrivial (does not have a fixed point) and minimal (that is X has no proper G-invariant subtrees). Let > 0 be an arbitrary real number. Then any finite generating set Y of G with k elements is Nielsen-equivalent to a set S such that:

(1) There is a ﬁnite subtree Y ⊆ X of measure at most 2M(k − 1) + such that GY = X; (2) for some x ∈ Y we have max{d(x, sx)|s ∈ S} ≤ 2M(k − 1) + .

By the measure of Y we mean the sum of the lengths of intervals in any subdivision of Y as a disjoint union of ﬁnitely many intervals. This is equal to the 1-dimensional Hausdorﬀ measure of Y . If X is a simplicial tree and Y is a simplicial subtree, then the measure of Y is the number of edges in Y. ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 3

Theorem 1.2 immediately implies the following since actions with trivial arc stabilizers are 0-acylindrical: Corollary 1.3. Let G be a finitely generated freely indecomposable non- elementary group which acts by isometries on an R-tree X with trivial arc stabilizers. Then for any > 0 and any finite generating set Y of G there is a set S Nielsen-equivalent to Y and a point x ∈ X such that d(x, sx) ≤ for all s ∈ S. Thus if the action of G is 0-acylindrical, that is arc stabilizers are trivial, then a finite generating set of G can be made by Nielsen transformations to have arbitrarily small translation length. Not surprisingly our methods let us recover the same bound c(G, M) = 2M(k − 1) on the complexity of acylindrical accessibility splittings as the one given in [40]. The main ingredient is a theory of Nielsen methods for groups acting on hyperbolic spaces that we systematically developed in [29, 30]. This theory is analogous to Weidmann’s treatment of actions on simplicial trees [40], but the case of arbitrary hyperbolic spaces is technically much more complicated. Note that the proof of Theorem 1.2 completely avoids the Rips machinery for groups acting on R-trees [8, 27] and Theorem 1.2 makes no traditional stability assumptions about the action. Rather, we use the fact that an R-tree is δ-hyperbolic for any δ > 0, which allows us to make a limiting argument for δ tending to zero.

2. The main technical tool Our main tool is a technical result of [29] (see Theorem 2.4 below). It is motivated by the Kurosh subgroup theorem (see [34, 3]) for free products, which states that a subgroup of a free product ∗i∈I Ai is itself a free product of a free group and subgroups that are conjugate to subgroups of the factors Ai. First, we equip every non-trivial subgroup U ≤ Isom(X) with a U- invariant quasiconvex subset X(U) (see Section 4 for the precise definition). The set X(U) generalizes the definition of the subtree TU in the simplicial tree action case [40] (in that situation TU contains the minimal U-invariant subtree as well as the edges of the ambient tree which are fixed by some nontrivial element of U). For a group U acting nontrivially and acylindrically on an R-tree X the set X(U) turns out to be Hausdorff-close to the minimal U-invariant subtree. We generalize the notion of Nielsen equivalence as follows. The objects which correspond to the tuples of elements of G are the G-tuples: Definition 2.1 (G-tuple). Let G be a group. Let n ≥ 0, m ≥ 0 be integers such that m + n > 0. We will say that a tuple M = (U1,...,Un; H) is a G-tuple if Ui is a non-trivial subgroup of G m for i ∈ {1, . . . , n} and H = (h1, . . . , hm) ∈ G is a m-tuple of elements of G. We will denote M = U1 ∪ · · · ∪ Un ∪ {h1, . . . , hm} and call M the underlying set of M. Note that M is nonempty since m + n > 0. 4 I. KAPOVICH AND R.WEIDMANN

By analogy with the Kurosh subgroup theorem we will sometimes refer to the subgroups Ui as elliptic subgroups of M. This is justiﬁed since in most applications of our methods (in particular the proof of Theorem 1.2) the subgroups Ui are generated by sets of elements with short translation length. We will also refer to H as the hyperbolic component of M. We have the following notion of equivalence for G-tuples which generalizes the classical Nielsen equivalence. Deﬁnition 2.2 (Equivalence of G-tuples). We will say that two G-tuples 0 0 0 0 M = (U1,...,Un; H) and M = (U1,...,Un; H ) are equivalent if H = 0 0 0 0 (h1, . . . , hm) and H = (h1, . . . , hm) and M can be obtained from M by a chain of moves of the following type: −1 (1) For some 1 ≤ j ≤ n replace Uj by gUjg where

g ∈ h{h1, . . . , hm} ∪ U1 ∪ ... ∪ Uj−1 ∪ Uj+1 ∪ ... ∪ Uni. 0 (2) For some 1 ≤ i ≤ n replace hi by hi = g1hig2 where

g1, g2 ∈ h{h1, . . . , hi−1, hi+1, . . . , hm} ∪ U1 ∪ ... ∪ Uni. Deﬁnition 2.3 (G-space). Let G be a group acting on a metric space (X, d) by isometries. In this case we will term (X, d) together with this action a G-space. We can now formulate the main technical tool used in this paper that we obtained in [30] : Theorem 2.4 (Kapovich-Weidmann). [30] For any integer k ≥ 1 there is a constant K = K(k) with the following property. Suppose (X, d) is a δ-hyperbolic strongly geodesic G-space. Let M = (U1,...,Un; H) be a G-tuple where H = (h1, . . . , hm), n + m ≤ k and n [ U = hMi = h{h1, . . . , hm} ∪ Uii ≤ G. i=1

Then either U = U1 ∗ ...Un ∗ F (H) or after replacing M by an equivalent 0 0 0 0 0 0 0 G-tuple M = (U1,...,Un; H ) with H = (h1, . . . , hm) one of the following holds: 0 0 (1) d(X(Ui ),X(Uj)) < δK for some 1 ≤ i < j ≤ n. 0 0 (2) d(X(Ui ), hjX(Ui )) < δK for some i ∈ {1, . . . , n} and j ∈ {1, . . . , m}. (3) There exists a x ∈ X such that d(x, hjx) < δK for some j ∈ {1, . . . , m}. The notion of ”strongly geodesic” hyperbolic space (see Definition 3.2) in the above theorem is a technical condition which insures that all possible definitions of the boundary of X coincide and that any two points in X ∪∂X can be connected by a geodesic. In particular, proper hyperbolic spaces, R- trees and complete CAT (−1)-spaces are strongly geodesic. Theorem 2.4 is used to obtain the main result. We deploy Theorem 2.4 for a ”generator transfer” process to analyze a freely indecomposable subgroup generated by a finite set Y = {y1, . . . , ym} with m elements. First we start ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 5 with a G-tuple M1 = (; HY ) where HY = (y1, . . . , ym). We then construct a sequence of G-tuples M1,M2,... by repeatedly applying Theorem 2.4 in order to either ”drag” elements of the ”hyperbolic” component of Mi into the ”elliptic” components of our G-tuples or to join two elliptic subgroups to one new elliptic subgroup. A simple observation shows that the length of the sequence M1,M2,... is bounded by 2m − 1. The desired result is then obtained by analyzing the terminal member of this sequence.

3. Hyperbolic metric spaces We will give only a quick overview of the main definitions related to Gromov-hyperbolic spaces. For the detailed background information the reader is referred to [24], [16], [25], [1], [15], [4]. A geodesic segment in a metric space (X, d) is an isometric embedding γ : [0, s] −→ X, where [0, s] ⊆ R. Similarly, a geodesic ray and a biinfinite geodesic are defined as isometric embeddings γ : [0, ∞) −→ X and γ : (−∞, ∞) −→ X accordingly. A metric space (X, d) is said to be geodesic if any two points in X can be joined by a geodesic segment. We will often denote a geodesic segment connecting a ∈ X to b ∈ X by [a, b] and identify this geodesic segment with its image in X. By a K-neighborhood of a set A ⊆ X we will always mean the closed K-neighborhood, that is {y ∈ X|d(y, A) ≤ K}. Two sets A, B ⊆ X are said to be K-Hausdorff close if A is contained in the K-neighborhood of B and B is contained in the K-neighborhood of A. If A, B ⊆ X are K-Hausdorff close for some K ≥ 0, they are said to be Hausdorff close. Two paths γ : I −→ X and γ0 : J −→ X are said to be K-Hausdorff close (where I,J are sub-intervals of the real line) if their images γ(I) and γ0(J) are K-Hausdorff close. Similarly, γ, γ0 are said to be Hausdorff-close if they are K-Hausdorff close for some K ≥ 0. We recall the definition of a δ-hyperbolic metric space. Definition 3.1 (Hyperbolic space). A geodesic metric space (X, d) is said to be δ-hyperbolic if for any geodesic triangle in X with geodesic sides α = [x, y], β = [x, z], γ = [y, z] and vertices x, y, z ∈ X each side of the triangle is contained in the δ-neighborhood of the union of two other sides. That is for any p ∈ α there is q ∈ β ∪ γ such that d(p, q) ≤ δ. A geodesic metric space is called hyperbolic if it is δ-hyperbolic for some δ ≥ 0. If (X, d) is a metric space and x, y, z ∈ X, one defines the Gromov product (y, z)x as 1 (y, z) := (d(y, x) + d(z, x) − d(y, z)). x 2 If (X, d) is a δ-hyperbolic space then the Gromov product measures for how long two geodesics stay close together. Namely, the initial segments of length (y, z)x of any two geodesics [x, y] and [x, z] in X are 2δ-Hausdorff close. One can attach to a hyperbolic space X with base-point a a ”space at infinity”, called the boundary of X and denoted ∂aX. The boundary is 6 I. KAPOVICH AND R.WEIDMANN usually defined as a set of equivalence classes of sequences in X convergent at infinity. Recall that a sequence (xn)n≥1 in X is convergent at infinity if lim supn,m→∞(xn, xm)a = ∞. Two such sequences (xn)n and (yn)n are equivalent if lim supn,m→∞(xn, ym)a = ∞. This means that for large n, m the segments [a, xn] and [a, ym] are δ-close for a long time. The boundary ∂aX is defined as the set of equivalence classes of sequences in X convergent at infinity. The equivalence class of (xn)n is denoted by [(xn)n]. The boundary is topologized by saying that [(xn)n] ∈ ∂aX and [(yn)n] ∈ ∂aX are close if lim supn,m→∞(xn, ym)a is large. One can extend this topology to a topology on X = X ∪ ∂aX by by saying that a point x ∈ X and a point at infinity [(xn)n] are close in X if lim supn→∞(x, xn)a is large, that is the segments [a, x] and [a, xn] are δ-close for a long time for large n. It is easy to see and well-known that the definition of the set ∂aX, and the topology on X = X ∪ ∂aX does not depend on the choice of a. Thus from now on we will suppress reference to the base-point. For a point x ∈ X and a point at infinity p = [(xn)n] we say that a geodesic ray γ in X with γ(0) = x connects x to p if p = [(xn)n] = [(γ(n))n]. Similarly, we say that a bi-infinite geodesic γ :(−∞, ∞) → X connects p1 = [(xn)n] ∈ ∂X to p2 = [(ym)m] ∈ ∂X if p1 = [(γ(−n))n] and p2 = [(γ(n))n]. Definition 3.2 (Strongly geodesic). We say that a hyperbolic geodesic metric space (X, d) is strongly geodesic if for any two distinct points in

X = X ∪ ∂x0 X can be connected by a geodesic in X. We refer the reader to [1, 4, 12, 13, 15, 16, 25, 24, 28] for detailed background information on boundaries of hyperbolic spaces. In the remainder of this article we will require all hyperbolic spaces to be strongly geodesic. Recall that a metric space is called proper if all closed metric balls are compact. It is well known that proper hyperbolic metric spaces [16, 25, 1] and complete CAT (−1)-spaces [2, 14] as well as R-trees are strongly geodesic. Moreover, for a proper hyperbolic space X both ∂X and X ∪ ∂X are compact. Deﬁnition 3.3 (Quasiconvexity). A subset A of a geodesic metric space (X, d) is said to be -quasiconvex in X (where ≥ 0) if any geodesic segment joining two points of A is contained in the -neighborhood of A. A subset A ⊆ X is called quasiconvex if it is -quasiconvex for some ≥ 0. Let X be a strongly geodesic hyperbolic metric space and let A ⊆ X ∪∂X be a nonempty subset. If A has just one element, we put Conv(A) = A. If A contains at least two elements we deﬁne Conv(A) to be the union of all geodesics joining distinct points of A. This set Conv(A) is termed the convex hull of A in X.

4. Limit sets and convex hulls Till the end of this section, unless specified otherwise, we assume that (X, d) is a strongly geodesic δ-hyperbolic G-space. In this section we associate to any non-trivial subgroup U ≤ G a set XU which roughly corresponds ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 7 to the minimal invariant subtree if X is a tree and a set X(U) which corresponds to the tree TU of [40]. Let x ∈ X. We define the limit set of U, denoted Λ(U), to be the collection of all p ∈ ∂X such that p = limn→∞ unx for some sequence un ∈ U. If y ∈ X is a different point then the orbits Ux and Uy are d(x, y)-Hausdorff close. Therefore the definition of Λ(U) does not depend on the choice of x ∈ X. We can now define an analogue of the minimal U-invariant subtree: Definition 4.1. Let U ≤ G be a non-trivial subgroup such that Λ(U) has at least two distinct points. Then we define the weak convex hull XU of U as XU := Conv(Λ(U)). Note that the assumption on the limit set is also necessary in the case when X is a real tree. Indeed a group U acting on a real tree by isometries either contains a hyperbolic element and has two distinct points in the limit set or the group acts with a fixed point in which case the minimal U-invariant subtree is not necessarily unique. Traditionally the weak convex hull XU is termed the ”convex hull” of U (see for example [28, 39]). It is a very useful and natural geometric object with many interesting applications. However, as in [40], it turns out that XU is not the right object for our purposes and we need to consider a bigger U-invariant set. Namely, if U acts on a simplicial tree T , it is necessary (see [40]) to study the set that contains not only the minimal U-invariant subtree but also the points that are fixed under the action of some non-trivial element of U. Following this analogy, we introduce the following notions: Definition 4.2. Let U ≤ G be a non-trivial subgroup. We define the small displacement set of U, denoted E(U), as E(U) := {x ∈ X|d(x, gx) ≤ 100δ for some g ∈ G, g 6= 1} We put Z(U) := Conv(Λ(U) ∪ E(U)). We now define X(U) := Conv(Z(U)) and call X(U) the convex hull of U. Recall that if g is an isometry of a metric space (X, d), the translation length ||g|| of g is defined as ||g|| = inf{d(x, gx)|x ∈ X}. We shall make use of the following simple lemma [30] which is particularly obvious in the case of R-trees since in that situation any isometry of X either fixes a point or acts by translation on a line in X: Lemma 4.3. [30] Let U ≤ G be a non-trivial subgroup. Then X(U) is nonempty. We summarize the properties of U-invariant subsets constructed above in the following two lemmas [30]: 8 I. KAPOVICH AND R.WEIDMANN

Lemma 4.4. [30] Let U ≤ G be a non-trivial subgroup and suppose that δ > 0 (Recall that δ is the hyperbolicity constant of X). Then

(1) The sets X(U), XU , E(U) and Z(U) are U-invariant. The set XU is 8δ-quasiconvex and the set X(U) is connected, closed and 4δ-quasiconvex. (2) For any g ∈ G we have E(gUg−1) = gE(U), Λ(gUg−1) = gΛ(U), −1 −1 Z(gUg ) = gZ(U), X(gUg ) = gX(U) and XgUg−1 = gXU . Lemma 4.5. [30] Let U ≤ G be a non-trivial subgroup and assume that δ > 0. Suppose A ⊆ X is a nonempty C-quasiconvex and U-invariant set such that (1) the set A is contained in the C-neighborhood of X(U); and (2) E(U) ⊆ A. Then X(U) and A are (3C + 4δ)-Hausdorﬀ close.

5. Groups acting on real trees Recall that an R-tree is a 0-hyperbolic geodesic metric space. Any R-tree is strongly geodesic and δ-hyperbolic for any δ > 0. We can give a more explicit description of the sets X(U),XU and E(U) for groups acting on R-trees. We deﬁne generating trees as in [10]: Deﬁnition 5.1. Let G be a group acting by isometries on an R-tree X and U = hSi be a subgroup. We say that a tree YS ⊂ X is a generating tree of U with respect to S if Y ∩ sY 6= ∅ for all s ∈ S. We further say that YU is a generating tree of U if YU is a generating tree for U with respect to some generating set S0 of U.

It is easy to see that if YS is a generating tree of U with respect to a generating set S of U then the set UYS is connected and U-invariant. Thus UYS contains the minimal U-invariant subtree of X, provided U does not have a global fixed point (see the lemma below). This observation justifies the term “generating tree”. Recall that a group U acting on an R-tree X either acts with a global fixed point or there exists a unique infinite minimal U-invariant subtree TU . In the latter case U contains hyperbolic elements, and hence the limit set ΛU is non-empty. The following is an immediate consequence of the above definitions: Lemma 5.2. Let X be an R-tree and let δ > 0 be an arbitrary number (so that X is δ-hyperbolic). Let U be a group acting on X by isometries. Suppose that U is generated by a finite set S and that U contains a hyperbolic element. Let X(U) be the convex hull of U (relative δ) and let TU ⊆ X be the minimal U-invariant subtree of X. Then

(1) TU = XU := Conv(Λ(U)) and therefore TU ⊆ X(U). (2) The set X(U) is connected and therefore a subtree of X. (3) For any generating tree YU the tree TU ∩ YU is a generating tree. (4) For any generating tree YU we have TU ⊂ UYU . ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 9

(5) If YU is a generating tree for U, then gYU is a generating tree for gUg−1. We can now observe that for acylindrical actions the “small displacement” set E(U) cannot be too far from the minimal U-invariant tree TU . Lemma 5.3. Let U be a group acting by isometries on an R-tree X and suppose this action is M-acylindrical for some M ≥ 0. Let TU be the minimal U-invariant subtree of X if U contains a hyperbolic element and let TU be any point of X ﬁxed by U otherwise. Then the following holds.

(1) Let y ∈ X be such that d(y, TU ) = R. Then for any u ∈ U, u 6= 1 we have d(y, uy) ≥ 2R − 2M. (2) Let δ > 0 be an arbitrary number and let X(U) and E(U) be the convex hull and the small displacement sets relative δ accordingly. Then the set E(U) is contained in the (50δ + M)-neighborhood of TU . Moreover, X(U) and TU are (50δ + M)-Hausdorﬀ close.

Proof. (1) Let x ∈ TU be such that d(y, x) = d(y, TU ) = R. Then ux ∈ TU and d(uy, ux) = d(uy, TU ) = R. Then for any t ∈ TU the path [y, x] ∪ [x, t] is a geodesic, since otherwise x would not be the closest point to y in TU . If ux 6= x then [y, x] ∪ [x, ux] ∪ [ux, uy] is a geodesic. Hence d(y, uy) ≥ 2R ≥ 2R − M, as required. Suppose now that ux = x. Let p ∈ X be such that [x, p] = [x, y] ∩ [x, uy]. Then [x, p] is point-wise fixed by u. Since by assumption u 6= 1 and the action is M-acylindrical, this means that d(x, p) ≤ M. We have d(y, p) = d(uy, up) = d(uy, p) ≥ R − M. The path [y, p] ∪ [p, ux] is a geodesic. Therefore d(y, uy) ≥ 2R − 2M, as required. (2) The definition of E(U) together with part (1) imply that E(U) is contained in the (50δ + M)-neighborhood of TU . We already know that TU ⊆ X(U), that Conv(ΛU) ⊆ TU and that TU and X(U) are subtrees of X. By definition X(U) = Conv(E(U) ∪ ΛU), which implies that X(U) is contained in the (50δ + M)-neighborhood of TU . Thus TU and X(U) are (50δ + M)- Hausdorff close. Theorem 5.4. Let M ≥ 0 be a real number. Let G be a freely indecomposable non-infinite cyclic finitely generated group acting by isometries on an R-tree X and suppose that the action is M-acylindrical. Suppose that G is generated by a tuple S = (s1, . . . , sk), where k ≥ 1. Then for any > 0 there exists a finite subtree YG ⊂ X of measure at most 2M(k − 1) + and a tuple S¯ = (¯s1,..., s¯k) that is Nielsen equivalent to S such that YG is a generating tree for G with respect to {s¯1,..., s¯k}. Before we give the proof of Theorem 5.4 we argue that it immediately implies Theorem 1.2.

Proof of Theorem 1.2. Let S = (s1, . . . , sk) be a generating tuple of G. It follows from Theorem 5.4 that there exists a generating tree YG of measure at most 2M(k−1)+ and a tuple S¯ = {s¯1,..., s¯k) which is Nielsen equivalent to S and such thats ¯iYG ∩ YG 6= ∅ for 1 ≤ i ≤ k. 10 I. KAPOVICH AND R.WEIDMANN

Since YG is a generating tree, GYG is a subtree of X that is G-invariant. The assumption that the action of G on X is minimal implies that GYG = X. Thus part (1) of Theorem 1.2 is established. Since YG is a tree, there exists a point x ∈ YG such that d(x, y) ≤ 1 2 (2M(k − 1) + ) for any point y ∈ YG. For 1 ≤ i ≤ k choose yi such 1 that yi ∈ YG ∩ s¯iYG. Then d(x, yi) ≤ 2 (2M(k − 1) + ) and d(¯six, yi) ≤ 1 2 (2M(k−1)+). The triangle inequality now yields d(x, s¯ix) ≤ 2M(k−1)+ for 1 ≤ i ≤ k which proves assertion (2) of Theorem 1.2. 2 Before we proceed with the proof of Theorem 5.4 let us recall some more notions from [30]. Let G be a group. If Y is an n-tuple of elements of G we will say that n is the length of Y which we denote by L(Y ). We will say that M = (Y1,...,Yp; H) is a partitioned tuple in G if p ≥ 0 and Y1,...,Yp,H are ﬁnite tuples of elements of G such that at least one of these tuples has positive length and such that for any i ≥ 1 the tuple Yi has positive length. We will call the sum of the lengths of L(Y1)+···+L(Yp)+L(H) the length of M and denote it by L(M). We further call the L(M)-tuple, obtained by concatenating the tuples Y1,...,Yp,H, the tuple underlying M.

Remark 5.5. To any partitioned tuple M = (Y1,...,Yp; H) we associate the G-tuple τ = (U1 = hY1i,...,Up = hYpi; H). Suppose now that τ is equivalent to a G-tupleτ ¯ = (U¯1,..., U¯p, H¯ ). The deﬁnition of equivalence of G-tuples implies that there is a partitioned tuple M¯ = (Y¯1,..., Y¯p; H¯ ) with associated G-tupleτ ¯ such that the tuples undelying M and M¯ are Nielsen-equivalent. Moreover, we can choose Y¯i to be conjugate to Yi for each 1 ≤ i ≤ p.

Proof of Theorem 5.4. We will prove Theorem 5.4 by induction on k. Sup- pose first k = 1, so that G = hs1i. Since by assumption G is not infinite cyclic, the element s1 has finite order and so G is a finite cyclic group. Therefore G fixes some point x ∈ X. Then YG := {x} is a generating tree for G with respect to S = (s1) and the conclusion of Theorem 5.4 obviously holds. Suppose now k > 1 and that Theorem 5.4 has been proved for all smaller values of k. Suppose S = (s1, . . . , sk) is a generating set of G. If S is Nielsen- equivalent to some tuple containing an entry equal 1, the statement of Theo- rem 5.4 follows from the induction hypothesis. Thus we may assume that for any k-tuple Nielsen-equivalent to S the entries of this tuple are non-trivial and pairwise distinct. Set δ := , K(k)(2k − 1) + 100(k − 1) where K(k) is the constant provided by Theorem 2.4. ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 11

The space X is δ-hyperbolic since X is an R-tree. We will show that there exists a generating tree YG of measure at most 2M(k − 1) + (K(k)(2k − 1) + 100(k − 1))δ = 2M(k − 1) + , which is the conclusion of Theorem 5.4. Let N = (S1,...Sn; H) be a partitioned tuple of elements of G. We say that N is good if Ui = hSii 6= 1 for all i ≥ 1 and if for each Ui, i ≥ 1 there exists a generating tree Yi of measure at most 2M(ki −1)+(K(k)(2ki −1)+ 100(ki − 1))δ, where ki = L(Si). We define N1 to be the partitioned tuple N1 = (; S). Clearly N1 is good. As in [40, 29], we define the complexity of a partitioned tuple M = 2 (S1,...,Sn; H) with H = (h1, . . . , hm) to be the pair (m, n) ∈ N (we as- 2 0 0 sume that 0 ∈ N). We define an order on N by setting (m, n) ≤ (m , n ) if 0 0 2 m < m or if m = m and n ≤ n . This gives a well-ordering on N .

Choice of M. Let M = (S1,...Sn; H) with H = (h1, . . . , hm) be a partitioned tuple of minimal complexity among all partitioned good tuples with the underlying tuple being Nielsen equivalent to S. (The partitioned tuple N1 satisﬁes the above qualifying constraints and hence such an M exists.) Thus M generates G and m + n ≤ k.

We will show that M = (S1; −) which immediately implies the assertion of Theorem 5.4. Suppose that M is not of this type. Recall that G is freely indecomposable and not inﬁnite cyclic. It follows from Theorem 2.4 and Remark 5.5 that we can replace M by a good partitioned tuple M¯ = (S¯1,... S¯n, H¯ ) of the same complexity as M with H¯ = (h¯1,..., h¯m) such that the underlying tuple of M¯ is Nielsen equivalent to S after a conjugation and such that the following holds. If we denote U¯i = hS¯ii for 1 ≤ i ≤ n then at least one of the following occurs:

(1) d(X(U¯1),X(U¯2)) ≤ δK(k); (2) d(X(U¯1), h¯mX(U¯1)) ≤ δK(k); (3) d(y, h¯my) ≤ δK(k) for some y ∈ X. Note that since both M and M¯ generate G, they are in fact Nielsen- equivalent and not simply Nielsen-equivalent after a conjugation. Denote ki = L(Si) for 1 ≤ i ≤ n.

(1) Suppose that d(X(U¯1),X(U¯2)) ≤ δK(k). Choose x1 ∈ X(U¯1) and x2 ∈ X(U¯2) such that d(x1, x2) ≤ δK(k). By part (2) of Lemma 5.3 there is y ∈ T such that d(y , x ) ≤ 50δ +M for i = 1, 2. It follows that d(y , y ) ≤ i U¯i i i 1 2 100δ + 2M + δK(k). Since by assumption M¯ is good, we can choose a generating tree Y for U¯ of measure at most 2M(k − 1) + (K(k)(2k − 1) + U¯i i i i 100(k − 1))δ. Since T ⊂ U¯ Y , there exists a u ∈ U¯ such that y ∈ u Y i U¯i i U¯i i i i i U¯i for i = 1, 2. Denote S0 := u S¯ u−1 for i = 1, 2. Then hS0i = hS¯ i = U¯ and u Y is the i i i i i i i i U¯i ¯ 0 generating tree for Ui with respect to the generating set Si. 0 ¯ ¯ Moreover Si is Nielsen-equivalent to Si since ui ∈ Ui for i = 1, 2. Put ¯ ¯ V := hU1, U2i. Then YV = u1YU¯1 ∪ [y1, y2] ∪ u2YU¯2 is a generating tree for 12 I. KAPOVICH AND R.WEIDMANN

0 0 V with respect to S1 ∪ S2. The measure of YV is at most

2M(k1 − 1) + (K(k)(2k1 − 1) + 100(k1 − 1))δ + 2M(k2 − 1)+

+ (K(k)(2k2 − 1) + 100(k2 − 1))δ + 100δ + 2M + δK(k) =

= 2M(k1 + k2 − 1) + (K(k)(2k1 + 2k2 − 1) + 100(k1 + k2 − 1))δ. 0 0 0 ¯ ¯ ¯ Hence the partitioned tuple M := (S1 ∪S2, S3,..., Sn; H) is good. Since the underlying tuple of M 0 is Nielsen equivalent to S and since M 0 has smaller complexity than M¯ , we obtain a contradiction with the choice of M.

(2) Suppose that d(X(U¯1), h¯mX(U¯1)) ≤ δK(k). As in (1) we see that ¯ there exist y1 ∈ TU¯1 and y2 ∈ hmTU¯1 such that d(y1, y2) ≤ 100δ + 2M + δK(k). Again, since M¯ is good, after replacing S1 by a Nielsen equivalent tuple we can assume that there exists a generating tree YU¯1 of measure at most 2M(k1 −1)+(K(k)(2k1 −1)+100(k1 −1))δ such that y1 ∈ YU¯1 . Clearly ¯−1 ¯ ¯−1 hm y2 lies in TU¯1 . Since U1YU¯1 = TU¯1 it follows that (¯u1hm )y2 ∈ YU¯1 for someu ¯1 ∈ U¯1.

Hence YV1 = YU¯1 ∪ [y1, y2] is a generating tree for the subgroup V1 = ¯ ¯−1 ¯ ¯ 0 ¯ ¯−1 hS1, u¯1hm i = hS1, hmi with respect to the generating set S1 = S1 ∪{u¯1hm }.

Moreover, the measure of YV1 is at most

2M(k1 − 1) + (K(k)(2k1 − 1) + 100(k1 − 1))δ + 100δ + 2M + δK(k) ≤

≤ 2Mk1 + (K(k)(2k1 + 1) + 100k1)δ. 0 0 0 0 This implies that the tuple M = (S1,S2,...,Sn; H ) is good, where H = 0 (h¯1,..., h¯m−1). Since M has smaller complexity than M, this contradicts the choice of M.

(3) Suppose that d(y, h¯my) ≤ δK(k) for some y ∈ X. In this case we replace M¯ by 0 0 M = (S¯1,..., S¯n, S¯n+1; H ) 0 where S¯n+1 = (h¯m) and H = (h¯1,..., h¯m−1). 0 By assumption on S we have hm 6= 1. Hence M is good and of smaller complexity than M, which again yields a contradiction. 2 As promised in the introduction, we recover a theorem of Weidmann [40] about acylindrical actions on simplicial trees: Theorem 5.6. Let G be a freely indecomposable group which is generated by k elements. Suppose that G is not inﬁnite cyclic. Let A be a minimal M- acylindrical graph of groups representing G. Then A has at most 2M(k − 1) edges. Proof. We only need to apply the fact that the measure of a approximating tree YG is at least as great as the measure of the graph Γ underlying the graph of groups. This holds since the set GYG is connected and G-invariant. It must therefore contain the minimal invariant subtree, see also [10] for ACYLINDRICAL ACCESSIBILITY FOR GROUPS ACTING ON R-TREES 13 details. Since for any > 0 there exists a generating tree of measure at most 2M(k − 1) + this implies that Γ has at most 2M(k − 1) edges.

References [1] J.Alonso, T.Brady, D.Cooper, V.Ferlini, M.Lustig, M.Mihalik, M.Shapiro and H.Short. Notes on hyperbolic groups. in “Group theory from a geometric viewpoint”, Proc. ICTP. Trieste, World Scientific, Singapore, 1991, 3–63 [2] W. Ballmann, Lectures on spaces of nonpositive curvature, BirkhäuserVerlag, Basel, 1995, With an appendix by Misha Brin. [3] H. Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. [4] N. Benakli and I. Kapovich, Boundaries of hyperbolic groups, “Combinato- rial and Geometric Group Theory” (R.Gilman et al, editors), Contemporary Mathematics, 296, pp. 39–93, American Mathematical Society, 2002 [5] M. Bestvina and M. Feighn, Bounding the complexity of simplicial group actions on trees. Invent. Math. 103 (1991), no. 3, 449–469 [6] M. Bestvina and M. Feighn, A counterexample to generalized accessibility. Ar- boreal group theory (Berkeley, CA, 1988), 133–141, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991 [7] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85–101. [8] M. Bestvina and M. Feighn, Stable actions of groups on real trees. Invent. Math. 121 (1995), no. 2, 287–321 [9] M. Bestvina and M. Feighn, Addendum and correction to: “A combination theorem for negatively curved groups” [J. Differential Geom. 35 (1992), no. 1, 85–101; MR 93d:53053], J. Differential Geom. 43 (1996), no. 4, 783–788. [10] O. Bogopolski and R. Weidmann On the uniqueness of factors of amalgamated products, J. Group Theory 5 (2002), no. 2, 233–240 [11] B. H. Bowditch, Notes on Gromov’s hyperbolicity criterion for path-metric spaces, in ”Group theory from a geometrical viewpoint (Trieste, 1990)”, 64– 167, World Sci. Publishing, 1991 [12] B. H. Bowditch, A topological characterisation of hyperbolic groups. J. Amer. Math. Soc. 11 (1998), no. 3, 643–667 [13] B. H. Bowditch, Boundaries of strongly accessible hyperbolic groups. The Ep- stein birthday schrift, 51–97 (electronic), Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry, 1998 [14] M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer- Verlag, Berlin, 1999. [15] J. W. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989), Oxford Univ. Press, New York, 1991, pp. 315–369. [16] M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrieet théoriedes groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics 111, Springer-Verlag, Berlin, 1990. [17] F. Dahmani, Combination of Convergence Groups, preprint, 2002 [18] T. Delzant, Sur l’accessibilite acylindrique des groupes de presentation finie, Ann. Inst. Fourier 49, No.4, 1215-1224 (1999). [19] T. Delzant and L. Potyagailo, Accessibilité hiérarchique des groupes de présentation finie, Topology 40 (2001), no. 3, 617–629 [20] M. Dunwoody, The accessibility of finitely presented groups. Invent. Math. 81 (1985), no. 3, 449–457 14 I. KAPOVICH AND R.WEIDMANN

[21] M. Dunwoody, An inaccessible group. Geometric group theory, Vol. 1 (Sussex, 1991), 75–78, London Math. Soc. Lecture Note Ser., 181, Cambridge Univ. Press, Cambridge, 1993 [22] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992 [23] R. Gitik, On the combination theorem for negatively curved groups, Internat. J. Algebra Comput. 6 (1996), no. 6, 751–760 [24] M. Gromov, Hyperbolic groups, Essays in group theory, editor S.M. Gersten, Springer-Verlag, MSRI Publications 8, 1985, 75–263. [25] E.´ Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’aprèsMikhael Gromov, BirkhäuserBoston Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. [26] I. Kapovich, The Combination Theorem and quasiconvexity, International Jour- nal of Algebra and Computation, 11 (2001), no. 2, 185–216 [27] M. Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathemat- ics, 183, Birkhäuser, Boston, 2001 [28] I. Kapovich and H. Short, Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups, Canad. J. Math. 48 (1996), no. 6, 1224–1244. [29] I. Kapovich and R. Weidmann, Nielsen Methods for groups acting on hyperbolic spaces, to appear in Geom. Dedicata [30] I.Kapovich and R.Weidmann, Freely indecomposable groups isometrically acting on hyperbolic spaces, to appear in Intern. J. Algebra and Comput. [31] I. Kapovich and D. Wise, The equivalence of some residual properties of word- hyperbolic groups. J. Algebra 223 (2000), no. 2, 562–583 [32] A. Karrass and D. Solitar, The free product of two groups with a malnormal amalgamated subgroup. Canad. J. Math. 23 (1971), 933–959 [33] O. Kharlampovich and A. Myasnikov, Hyperbolic groups and free constructions, Trans. Amer. Math. Soc. 350 (1998), no. 2, 571–613 [34] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer-Verlag, Berlin, 1977, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89. [35] M. Mitra, Height in splittings of hyperbolic groups, preprint, 1997 [36] Z.Sela, The isomorphism problem for hyperbolic groups. I. Ann. of Math. (2) 141 (1995), no. 2, 217–283 [37] Z. Sela, Acylindrical Accessibility, Invent. Math. 129, 1997, 527-565. [38] Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II, Geom. Funct. Anal. 7 (1997), no. 3, 561–593 [39] Z. Sela, Quasi-convex groups of isometries of negatively curved spaces, Topol. Appl. 110 (2001), no. 1, 119–129. [40] R. Weidmann The Nielsen method for groups acting on trees, Proc. London Math. Soc. 85 (2002)

Dept. of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA E-mail address: [email protected]

Fakultat¨ fur¨ Mathematik, Ruhr-Universitat¨ Bochum, Germany E-mail address: [email protected]