JournalsggdVol. 14, No. 3pp. 765–790

Finitely F\mathcal{F}-amenable actions and decomposition complexity of groups

  • Andrew Nicas

    McMaster University, Hamilton, Canada
  • David Rosenthal

    St. John's University, Queens, USA
Finitely $\mathcal{F}$-amenable actions and decomposition complexity of groups cover
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Abstract

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

DOI 10.4171/GGD/562