Growth tight actions of product groups

  • Christopher H. Cashen

    University of Vienna, Wien, Austria
  • Jing Tao

    University of Oklahoma, Norman, USA


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 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 metric on the product space, for all . In particular, the 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