Finitely $\mathcal{F}$-amenable actions and decomposition complexity of groups

Abstract

In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $\mathcal F$-amenable" group action, where $\mathcal F$ is a family of subgroups. We show how a finitely $\mathcal F$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $\mathcal F$ are bounded from above, gives an upper bound for the asymptotic dimension of $G$ viewed as a metric space with a proper left invariant metric. We generalize this to families $\mathcal F$ whose elements are contained in a collection, $\mathcal C$, of metric families that satisfies some basic permanence properties: If $G$ is a countable group and each element of $\mathcal F$ belongs to $\mathcal C$ and there exists a finitely $\mathcal F$-amenable action of $G$ on a compact metrizable space, then $G$ is in $\mathcal C$. Examples of such collections of metric families include: metric families with weak finite decomposition complexity, exact metric families, and metric families that coarsely embed into Hilbert space.