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

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.