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

In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the\nconcept of a "finitely $\\mathcal{F}$-amenable" group action, where\n$\\mathcal{F}$ is a family of subgroups. We show how a finitely\n$\\mathcal{F}$-amenable action of a countable group $G$ on a compact metric\nspace, where the asymptotic dimensions of the elements of $\\mathcal{F}$ are\nbounded from above, gives an upper bound for the asymptotic dimension of $G$\nviewed as a metric space with a proper left invariant metric. We generalize\nthis to families $\\mathcal{F}$ whose elements are contained in a collection,\n$\\mathfrak{C}$, of metric families that satisfies some basic permanence\nproperties: If $G$ is a countable group and each element of $\\mathcal{F}$\nbelongs to $\\mathfrak{C}$ and there exists a finitely $\\mathcal{F}$-amenable\naction of $G$ on a compact metrizable space, then $G$ is in $\\mathfrak{C}$.\nExamples of such collections of metric families include: metric families with\nweak finite decomposition complexity, exact metric families, and metric\nfamilies that coarsely embed into Hilbert space.\n

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