ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 2, Number 2, October 1996 S 1079-6762(96)00013-3

ON THE CUT POINT CONJECTURE

G. A. SWARUP

(Communicated by Walter Neumann)

Dedicated to John Stallings on his 60th birthday

Abstract. We sketch a proof of the fact that the Gromov boundary of a hyperbolic group does not have a global cut point if it is connected. This implies, by a theorem of Bestvina and Mess, that the boundary is locally connected if it is connected.

The object of this note is to sketch a proof of the following theorem:

Theorem 1. If Γ is a one ended hyperbolic group, then the Gromov boundary ∂Γ of Γ does not have a global cut point.

It seems that several people independently thought of using treelike structures in this context, among them Bill Grosso in an unpublished manuscript [8]. The most significant advances in this direction were carried out by Brian Bowditch in a brilliant series of papers ([4]-[7]). We draw heavily from his work. He proved Theorem 1 in the case when Γ is one ended and does not split over a two ended group [6], and in the case when Γ is strongly accessible and one ended [5]. Our strategy is to relativize some of the arguments of [4], [5], use Levitt’s construction [9] to obtain an R-tree on which a subgroup of Γ acts isometrically, and then use a relative version of the main theorem of [2] to arrive at a contradiction. One of the main results of [4] asserts:

Theorem 2 (Bowditch [4]). If ∂Γ has a global cut point, then ∂Γ has a nontrivial, equivariant dendrite quotient D(∂Γ) on which Γ acts as a discrete convergence group.

We refer to [4] for definitions; a dendrite is a compact separable R-tree. We assume that ∂Γ has a global cut point, and we start with a graph of group decomposition of Γ over two ended subgroups in which none of the vertex groups splits over a finite or two ended subgroup relative to the edge groups in it. We may assume that the action of Γ on the associated tree Σ is minimal, reduced, without inversions, and Σ/Γ is compact. Since we are assuming that ∂Γ has a global cut point, such splittings of Γ exist by [6] and [1].

\begin{matrix}c \\ ◯ \end{matrix}1996 American Mathematical Society

Received by the editors June 4, 1996.

1991 Mathematics Subject Classification. Primary 20F32; Secondary 20J05, 57M40.

Key words and phrases. Gromov hyperbolic group, Gromov boundary, cut point, local connectedness, dendrite, R-tree.

By [5], [7], there is a natural decomposition ∂Γ = ∂_{0}Γ ∪ ∂_{∞}Γ, where

∂_{0}Γ = ⋃_{v∈V (Σ)} ΛΓ(v)

and ∂_{∞}Γ is naturally identified with ∂Σ (ΛG for a subgroup of Γ denotes the limit set of G in ∂Γ and may be identified with ∂G if G is quasiconvex in Γ). Our starting point was the observation that the end points of an edge group cannot be separated by a global cut point, and it follows from the description in [4] of the quotient map ∂Γ → D(∂Γ) that:

Proposition 3. The fixed points of Γ(e), for each edge e, are identified in the dendrite quotient D(∂Γ).

This was first observed by the methods of [10] and the generalized accessibility of [1], but Bowditch pointed out that it follows immediately from Proposition 4 below. We now consider the images in D(∂Γ) of Λ(Γ(v), for v ∈ V (Σ). Bowditch considers for each directed edge e⃗ of Σ a subset Ψ(e⃗) of ∂Γ defined as follows. Let Φ(e⃗) denote the connected component of Σ minus the interior of e which contains the tail of e⃗. Then, Ψ(e⃗) is the part of the limit set of Γ to the left of e⃗:

Ψ(e⃗) = ∂Φ(e⃗) ∪ ⋃_{v∈V (Φ(e⃗))} ΛΓ(v).

Let Δ⃗(v) denote the set of directed edges of Σ with head at v.

Proposition 4 (Bowditch, Lemma 3.3 and Proposition 5.1 of [5]). With the notation above, we have

Ψ(e⃗) ∪ Ψ(−e⃗) = ∂Γ = ΛΓ(v) ∪ ⋃_{e⃗∈Δ⃗(v)} Ψ(e⃗),

Ψ(e⃗) ∩ Ψ(−e⃗) = ∂Γ(e),

and Ψ(e⃗) is closed, Γ(e) invariant, and connected.

The fact that Ψ(e⃗) is connected requires some argument. Since Ψ(e⃗) and ∂Γ are connected, it follows that by identifying the two points in ΛΓ(e) for e ∋ v we obtain a connected quotient of ∂Γ(v). This together with Proposition 3 shows that the image of ΛΓ(v) in D(∂Γ) is connected for each vertex v of Σ. Denote this by D(M(v)). At least one of these must be nontrivial (not a point) from the decomposition of ∂Γ described before Proposition 3. Choose one such vertex v; we denote the quotient D(M(v)) by D(M), Γ(v) by G, and call Γ(e) for e ∋ v the peripheral subgroups of G. Restricting the action of Γ to G we see that:

Proposition 5. G acts as a discrete convergence group on D(M), and the peripheral subgroups of G fix points of M. Moreover, D(M) is a nontrivial dendrite.

By removing the terminal points of D(M) one obtains as in [4] an R-tree T on which the induced action of G is nonnesting (see Section 7 of [4]; nonnesting means that for any compact interval A, if g(A) ⊂ A, then g(A) = A). Since the action of G on D(M) is a discrete convergence group action, the edge stabilizers for the action on T are finite. A peripheral subgroup H of G fixes a point of D(M) and thus fixes either a point of T or exactly one end of T. Since a nonnesting action on a real tree does not have parabolic elements (in the usual sense; see [9]), it follows that the peripheral subgroups of G fix points of T. In summary, we have:

Proposition 6. If ∂Γ has a global cut point, then some vertex group Γ(v) = G admits a nonnesting action on an R-tree T with finite edge stabilizers and such that each peripheral subgroup of G fixes some point of T.

Under these conditions Levitt ([9], see also [6] — perhaps it is possible to use [6] too for the rest of the argument) constructs an action of G by isometries on an R-tree T_{0} such that a subgroup of G fixing an arc of T_{0} also fixes an arc of T. The construction is natural, and since we have only finitely many conjugacy classes of peripheral subgroups, we see that the peripheral subgroups of G still fix points of T_{0} (this needs enlarging K in Levitt’s construction; see Corollary 6 of [9]). Thus we have the analogue of Proposition 6 with the action of G by isometries rather than just homeomorphisms. The condition on edge stabilizers shows that the action is stable in the sense of Bestvina and Feighn [2]. Thus by the relative version of the main theorem of [2] (this is Theorem 9.6 of [2]), we conclude that there is a nontrivial decomposition of G along a virtually cyclic group such that each peripheral subgroup is conjugate to a subgroup of a vertex group. Since we started with a G = Γ(v) which does not admit such a splitting, we have the desired contradiction to complete the proof of Theorem 1. By [3], it follows that ∂Γ is locally connected and the theory of [7] goes through for all hyperbolic groups. This leads to a precise characterization of when Out(Γ) is infinite.

Acknowledgement

I am grateful to Mladen Bestvina, Brian Bowditch, Mark Feighn, Gilbert Levitt, and Chuck Miller for their help in the preparation of this note. Of course, it would not have been possible to write this note without Bowditch’s outstanding work.

References

[1] M. Bestvina and M. Feighn, Bounding the complexity of simplicial actions on trees, Inv. Math. 103 (1993), 449–469. MR 92c:20044

[2] , Stable actions of groups on real trees, Inv. Math. 121 (1995), 287-361. MR 96h:20056

[3] M. Bestvina and G. Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), 469–481. MR 93j:20076

[4] B. H. Bowditch, Treelike structures arising from continua and convergence groups, Preprint (1995).

[5] , Boundaries of strongly accessible groups, Preprint (1996).

[6] , Group actions on trees and dendrons, Preprint (1995).

[7] , Cut points and canonical splittings of hyperbolic groups, Preprint (1995).

[8] W. Grosso, Unpublished manuscript, Berkeley (1995).

[9] G. Levitt, Nonnesting actions on real trees, Preprint (1996).

[10] G. P. Scott and G. A. Swarup, An algebraic annulus theorem, Preprint (1995).

The University of Melbourne, Parkville, 3052, Victoria, Australia

   E-mail address: gadde@maths.mu.oz.au