Acylindrical accessibility for groups acting on ?-trees

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 D-acylindrically on an R-tree X then there is a finite subtree T ε ⊆ X of measure at most 2D(k − 1) + ε such that GT ε = X. This generalizes theorems of Z. Sela and T. Delzant about actions on simplicial trees. (2000): 20F67 Mathematics Subject Classification

Freely Indecomposable Groups Acting on Hyperbolic Spaces

“…This statement is a close analogue of the main result of [43]. This statement turns out to also have applications to 3-manifold groups and group actions on real and simplicial trees [33]. Theorem 1.7.…”

Entropy and finiteness of groups with acylindrical splittings

“…The most elementary examples of groups possessing a k-acylindrical splitting are the free products and the fundamental groups of compact surfaces of negative Euler characteristic, but the class is considerably larger and encompasses several interesting classes of amalgamated groups which naturally arise in Riemannian and metric geometry, as we shall see. During the years, the existence of acylindrical actions on simplicial trees has been mainly used to prove some accessibility results ( [59] and [34]). More recently, there has been an increasing interest on groups acting acylindrically on Gromov hyperbolic spaces (see [50], [45], [61], and references therein for this more general notion of acylindricity).…”

On purely loxodromic actions

“…Definition 1.2 (Acylidrical actions). An isometric action of a group G on a Gromov-hyperbolic space X is said to be acylindrical if for every R ≥ 0 there exist L ≥ 1 and M ≥ 1 such that whenever x, y ∈ X are such that d X (x, y) ≥ L then # ({g ∈ G|d X (x, gx) ≤ R, d X (y, gy) ≤ R}) ≤ M Acylidrical actions on hyperbolic spaces play a crucial role in studying various generalizations of relatively hyperbolic groups, particularly the so-called acylindrically hyperbolic groups (see, for example [9,32,19,15,33]), and in the study of group actions on R-trees (see, for example, [10,21,34,1]). The action of M od(S) on the curve complex C(S) is also known to be acylindrical, see [6] and this fact has many useful consequences in the study of mapping class groups.…”