In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely -amenable" group action, where is a family of subgroups. We show how a finitely -amenable action of a countable group on a compact metric space, where the asymptotic dimensions of the elements of are bounded from above, gives an upper bound for the asymptotic dimension of viewed as a metric space with a proper left invariant metric. We generalize this to families whose elements are contained in a collection, , of metric families that satisfies some basic permanence properties: If is a countable group and each element of belongs to and there exists a finitely -amenable action of on a compact metrizable space, then is in . 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.
Cite this article
Andrew Nicas, David Rosenthal, Finitely -amenable actions and decomposition complexity of groups. Groups Geom. Dyn. 14 (2020), no. 3, pp. 765–790DOI 10.4171/GGD/562