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