Equivariant dimensions of groups with operators

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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.
Original language English Groups, Geometry and Dynamics Accepted/In press - 23 Dec 2020

Keywords

• 55N91
• 20J05 (Primary)
• 55M30
• 20E36 (Secondary)
• equivariant group cohomology
• equivariant Lusternik–Schnirelmann category
• classifying spaces