2,425,319 research outputs found
Tensor-product coaction functors
For a discrete group , we develop a `-balanced tensor product' of two
coactions and , which takes place on a certain
subalgebra of the maximal tensor product . Our motivation
for this is that we are able to prove that given two actions of , the dual
coaction on the crossed product of the maximal-tensor-product action is
isomorphic to the -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 , 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 is the action by translation on
, 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
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 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
- β¦