Equivariant dimensions of groups with operators

TL;DR

This paper introduces equivariant cohomological, geometric, and Lusternik-Schnirelmann dimensions for groups with automorphism actions, extending classical theorems to the equivariant setting and establishing their equivalence under certain conditions.

Contribution

It defines new equivariant invariants for groups with automorphisms and extends classical theorems to the equivariant context, showing their equivalence when the acting group is finite.

Findings

All three invariants coincide for finite G, except possibly in a specific case.

The cohomological dimension of finite groups relative to proper subgroup families exceeds one.

A Stallings-Swan type theorem is established for subgroup families not containing all finite subgroups.

Abstract

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\sf cd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and the equivariant Lusternik-Schnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with ${\sf cat}_G(\pi)={\sf cd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings-Swan type result for families of subgroups which do not contain all finite subgroups.