On equivariant asymptotic dimension (Q2406845)

Summary: 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 \textit{A. Mole} and \textit{H. Rüping} [``Equivariant refinements'', Preprint, \url{arXiv:1308.2799}] about equivariant refinements from finite groups to infinite groups.