Finitely $\mathcal{F}$-amenable actions and Decomposition Complexity of Groups

TL;DR

This paper explores how finitely -amenable actions of groups on compact spaces influence the asymptotic dimension of the group, extending the concept to broader families of metric spaces with certain properties.

Contribution

It generalizes the relationship between finitely -amenable actions and asymptotic dimension to families of metric spaces with permanence properties.

Findings

Finitely -amenable actions provide upper bounds on group asymptotic dimension.

The results apply to families with properties like finite decomposition complexity and coarse embeddability.

The framework unifies various classes of metric families under a common theoretical approach.

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, $\mathfrak{C}$, of metric families that satisfies some basic permanence properties: If $G$ is a countable group and each element of $\mathcal{F}$ belongs to $\mathfrak{C}$ and there exists a finitely $\mathcal{F}$-amenable action of $G$ on a compact metrizable space, then $G$ is in $\mathfrak{C}$. 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.