An Eilenberg-Ganea Phenomenon for Actions with Virtually Cyclic Stabilisers

TL;DR

This paper investigates the discrepancy between algebraic and geometric dimensions of classifying spaces for group actions with virtually cyclic stabilizers, revealing a phenomenon in Coxeter groups where these dimensions differ.

Contribution

It demonstrates a specific Eilenberg-Ganea type phenomenon for Coxeter groups with virtually cyclic stabilizers, showing a gap between Bredon cohomological and geometric dimensions.

Findings

Bredon cohomological dimension is 2

Bredon geometric dimension is 3

Phenomenon occurs in certain Coxeter groups

Abstract

In dimension 3 and above, Bredon cohomology gives an acurate purely algebraic description of the minimal dimension of the classifying space for actions of a group with stabilisers in any given family of subgroups. For some Coxeter groups and the family of virtually cyclic subgroups we show that the Bredon cohomological dimension is 2 while the Bredon geometric dimension is 3.