Groups, Geometry, and Dynamics

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Volume 11, Issue 3, 2017, pp. 977–1002
DOI: 10.4171/GGD/419

Published online: 2017-08-22

On equivariant asymptotic dimension

Damian Sawicki[1]

(1) Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

The work discusses equivariant asymptotic dimension (also known as "wide equivariant covers", "$N$-$\mathcal F$-amenability" or "amenability dimension" and "$d$-BLR condition") and its generalisation, transfer reducibility, which are versions of asymptotic dimension invented for the proofs of the Farrell–Jones and Borel conjectures.

We prove that groups of null equivariant asymptotic dimension are exactly virtually cyclic groups. We show that a covering of the boundary always extends to a covering of the whole compactification. We provide a number of characterisations of equivariant asymptotic dimension in the general setting of homotopy actions, including equivariant counterparts of classical characterisations of asymptotic dimension. Finally, we strengthen the result of Mole and Rüping about equivariant refinements from finite groups to infinite groups.

Keywords: Equivariant cover, asymptotic dimension, homotopy action, transfer reducible group, Farrell–Jones conjecture

Sawicki Damian: On equivariant asymptotic dimension. Groups Geom. Dyn. 11 (2017), 977-1002. doi: 10.4171/GGD/419