Hyperbolic Dimension and Decomposition Complexity

Nicas, Andrew; Rosenthal, David I.

doi:10.48550/arxiv.1509.06437

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Decomposition Over a Collection Of Metric Families1

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2018

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“…We conclude this section by stating three alternative definitions (established in [NR18]) for a metric family X to be n-decomposable over a collection of metric families C. Condition (C) is a key technical tool needed for the proofs of Theorems 3.7, 3.9 and 3.10.…”

Section: Decomposition Over a Collection Of Metric Familiesmentioning

confidence: 99%

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

Nicas¹,

Rosenthal²

2018

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In his work on the Farrell-Jones Conjecture, Arthur Bartels introduced the concept of a "finitely F -amenable" group action, where F is a family of subgroups. We show how a finitely F -amenable action of a countable group G on a compact metric space, where the asymptotic dimensions of the elements of 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 F whose elements are contained in a collection, C, of metric families that satisfies some basic permanence properties: If G is a countable group and each element of F belongs to C and there exists a finitely F -amenable action of G on a compact metrizable space, then G is in 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.

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Section: Decomposition Over a Collection Of Metric Familiesmentioning

confidence: 99%