Growth tight actions

Goulnara N. Arzhantseva, Christopher H. Cashen and Jing Tao

Vol. 278 (2015), No. 1, 1–49

DOI: 10.2140/pjm.2015.278.1

Abstract

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, nonelementary group $G$ acting on a $G$-space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting element and if $G$ is not too badly distorted in $\mathcal{X}$, then the action of $G$ on $\mathcal{X}$ is a growth tight action. It follows that if $\mathcal{X}$ is a cocompact, relatively hyperbolic $G$-space, then the action of $G$ on $\mathcal{X}$ is a growth tight action. This generalizes all previously known results for growth tightness of cocompact actions: every already known example of a group that admits a growth tight action and has some infinite, infinite index normal subgroups is relatively hyperbolic; conversely, relatively hyperbolic groups admit growth tight actions. This also allows us to prove that many CAT(0) groups, including flip-graph manifold groups and many right angled Artin groups, and snowflake groups admit cocompact, growth tight actions. These provide first examples of non relatively hyperbolic groups admitting interesting growth tight actions. Our main result applies as well to cusp uniform actions on hyperbolic spaces and to the action of the mapping class group on Teichmüller space with the Teichmüller metric. Towards the proof of our main result, we give equivalent characterizations of strongly contracting elements and produce new examples of group actions with strongly contracting elements.

Keywords

growth tight, Hopf property, entropy, relatively hyperbolic groups, mapping class groups, snowflake groups, rank 1 isometries, CAT(0) groups

Mathematical Subject Classification 2010

Primary: 20E06, 20F65, 20F67, 37C35, 57MXX

Milestones

Received: 20 February 2015

Accepted: 2 March 2015

Published: 30 September 2015