On equivariant asymptotic dimension

TL;DR

This paper investigates equivariant asymptotic dimension, establishing its connection to virtually cyclic groups, extending boundary coverings, and providing new characterisations and refinements for infinite groups.

Contribution

It proves that groups with null equivariant asymptotic dimension are exactly virtually cyclic, and extends boundary coverings and refinements to infinite groups.

Findings

Groups of null equivariant asymptotic dimension are exactly virtually cyclic groups.

Boundary coverings extend to entire compactifications.

Refinements from finite to infinite groups are established.

Abstract

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 Rueping (arXiv:1308.2799) about equivariant refinements from finite groups to infinite groups.