Counting problems in graph products and relatively hyperbolic groups
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COUNTING PROBLEMS IN GRAPH PRODUCTS AND RELATIVELY HYPERBOLIC GROUPS
BY
Ilya Gekhtman Department of Mathematics, University of Toronto 40 St George St., Toronto, ON, Canada e-mail: [email protected] AND
Samuel J. Taylor Department of Mathematics, Temple University 1805 North Broad St., Philadelphia, PA 19122, USA e-mail: [email protected] AND
Giulio Tiozzo Department of Mathematics, University of Toronto 40 St George St., Toronto, ON, Canada e-mail: [email protected]
ABSTRACT
We study properties of generic elements of groups of isometries of hyperbolic spaces. Under general combinatorial conditions, we prove that loxodromic elements are generic (i.e., they have full density with respect to counting in balls for the word metric in the Cayley graph) and translation length grows linearly. We provide applications to a large class of relatively hyperbolic groups and graph products, including all right-angled Artin groups and right-angled Coxeter groups.
Received October 13, 2018 and in revised form May 22, 2019
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I. GEKHTMAN, S. J. TAYLOR AND G. TIOZZO
Isr. J. Math.
1. Introduction Let G be a finitely generated group. One can learn a great deal about the geometric and algebraic structure of G by studying its actions on various negatively curved spaces. Indeed, Gromov’s theory of hyperbolic groups [Gro87] provides the clearest illustration of this philosophy. However, weaker forms of negative curvature, ranging from relative hyperbolicity [Far98, Bow12, Osi06] to acylindrical hyperbolicity [Osi16, Bow08], apply to much larger classes of groups and still provide rather strong consequences. In all of these theories, a special role is played by the loxodromic (or hyperbolic) elements of the action, i.e., those elements which act with sink-source dynamics. In this paper, we are interested in quantifying the abundance of such isometries for the action of G on a hyperbolic space X. We emphasize that in all but the simplest situations, the natural hyperbolic spaces that arise are not locally compact. This includes actions associated to relatively hyperbolic groups [Far98], cubulated groups [KK14, Hag14], mapping class groups [MM99], and Out(Fn ) [BF14, HM13], to name only a few. Hence, in this paper we make no assumptions of local finiteness or discreteness of the action. Suppose that G X is an action by isometries on a hyperbolic space X. We address the question: How does a typical element of G act on X? When G is not amenable, the word “typical” has no well defined meaning, and depends heavily on the averaging procedure: a family of finitely supported measures exhausting G. Although much is now known about measures generated from a random walk on G [Mah11, CM15, MT18, MS20], very little is known about counting with respect to balls in the word metric. This will be our main focus. In more precise terms, fix a finite generating set S for the group G. Let Bn be the ball of radius n about 1 with respect to the word metric d determined by S. Then we call a property P generic if #{g ∈ Bn : g has P } →1 #B
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