2018/04/24 by Andrew Nicas, Nicas, Andrew, David I. Rosenthal +1
Mathematics · #20F65 (Secondary) #20F69 (Primary) #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1804.09207
openalex publication_date 2018/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the\nconcept of a "finitely \F-amenable" group action, where\n\F is a family of subgroups. We show how a finitely\n\F-amenable action of a countable group G on a compact metric\nspace, where the asymptotic dimensions of the elements of \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 \F whose elements are contained in a collection,\n mathfrakC, of metric families that satisfies some basic permanence\nproperties: If G is a countable group and each element of \F\nbelongs to mathfrakC and there exists a finitely \F-amenable\naction of G on a compact metrizable space, then G is in mathfrakC.\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