# Growth tight actions of product groups

### Christopher H. Cashen

University of Vienna, Wien, Austria### Jing Tao

University of Oklahoma, Norman, USA

## Abstract

A group action on a metric space is called growth tight if the exponential growth rate of the group with respect to the induced pseudo-metric is strictly greater than that of its quotients. A prototypical example is the action of a free group on its Cayley graph with respect to a free generating set. More generally, with Arzhantseva we have shown that group actions with strongly contracting elements are growth tight.

Examples of non-growth tight actions are product groups acting on the $L_{1}$ products of Cayley graphs of the factors.

In this paper we consider actions of product groups on product spaces, where each factor group acts with a strongly contracting element on its respective factor space. We show that this action is growth tight with respect to the $L_{p}$ metric on the product space, for all $1<p≤∞$. In particular, the $L_{∞}$ metric on a product of Cayley graphs corresponds to a word metric on the product group. This gives the rst examples of groups that are growth tight with respect to an action on one of their Cayley graphs and non-growth tight with respect to an action on another, answering a question of Grigorchuk and de la Harpe.

## Cite this article

Christopher H. Cashen, Jing Tao, Growth tight actions of product groups. Groups Geom. Dyn. 10 (2016), no. 2, pp. 753–770

DOI 10.4171/GGD/364