GROWTH TIGHT ACTIONS OF PRODUCT GROUPS

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 L1 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 re-spective factor space. We show that this action is growth tight with respect to the Lp 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 first 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. 1