Subgroup posets, Bredon cohomology and equivariant Euler characteristics

TL;DR

This paper develops a new approach using Bredon cohomology and subgroup posets to analyze the geometric and algebraic properties of certain groups, providing formulas for Euler characteristics and dimensions of classifying spaces.

Contribution

It introduces a Bredon projective resolution based on subgroup posets, offering new insights into the dimensions of classifying spaces and Euler characteristics for virtually solvable groups.

Findings

Derived a Bredon projective resolution for groups with finiteness conditions.

Reproved that for virtually solvable groups, geometric and cohomological dimensions coincide.

Provided a formula for the Euler class of classifying spaces, enabling computation of orbifold Euler characteristics.

Abstract

For a discrete group $\Gamma$ satisfying some finiteness conditions we give a Bredon projective resolution of the trivial module in terms of projective covers of the chain complex associated to certain posets of subgroups. We use this to give new results on dimensions of $E\gamma$ and to reprove that for virtually solvable groups, $\underline{\cd}\Gamma=\vcd\Gamma$. We also deduce a formula to compute the Euler class of $E\gamma$ for $\Gamma$ virtually solvable of type $\FP_\infty$ and use it to compute orbifold Euler characteristics.