# Finitely $F$-amenable actions and decomposition complexity of groups

### Andrew Nicas

McMaster University, Hamilton, Canada### David Rosenthal

St. John's University, Queens, USA

## Abstract

In his work on the Farrell–Jones Conjecture, Arthur Bartels introduced the concept of a "finitely $F$-amenable" group action, where $F$ is a family of subgroups. We show how a finitely $F$-amenable action of a countable group $G$ on a compact metric space, where the asymptotic dimensions of the elements of $F$ 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 $F$ whose elements are contained in a collection, $C$, of metric families that satisfies some basic permanence properties: If $G$ is a countable group and each element of $F$ belongs to $C$ and there exists a finitely $F$-amenable action of $G$ on a compact metrizable space, then $G$ is in $C$. 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.

## Cite this article

Andrew Nicas, David Rosenthal, Finitely $F$-amenable actions and decomposition complexity of groups. Groups Geom. Dyn. 14 (2020), no. 3, pp. 765–790

DOI 10.4171/GGD/562