Tensor-product coaction functors

For a discrete group $G$, we develop a `$G$-balanced tensor product' of two coactions $(A,\delta)$ and $(B,\epsilon)$, which takes place on a certain subalgebra of the maximal tensor product $A\otimes_{\max} B$. Our motivation for this is that we are able to prove that given two actions of $G$, the dual coaction on the crossed product of the maximal-tensor-product action is isomorphic to the $G$-balanced tensor product of the dual coactions. In turn, our motivation for this is to give an analogue, for coaction functors, of a crossed-product functor originated by Baum, Guentner, and Willett, and further developed by Buss, Echterhoff, and Willett, that involves tensoring an action with a fixed action $(C,\gamma)$, then forming the image inside the crossed product of the maximal-tensor-product action. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces the tensor-crossed-product functor of Baum, Guentner, and Willett. We prove that every such tensor-product coaction functor is exact, thereby recovering the analogous result for the tensor-crossed-product functors of Baum, Guentner, and Willett. When $(C,\gamma)$ is the action by translation on $\ell^\infty(G)$, we prove that the associated tensor-product coaction functor is minimal, generalizing the analogous result of Buss, Echterhoff, and Willett for tensor-crossed-product functors.Comment: Minor revisio

Growth Tight Actions of Product Groups

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 $1<p\leq \infty$. In particular, the $L^\infty$ 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.Comment: 13 pages v2 15 pages, minor changes, to appear in Groups, Geometry, and Dynamic