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 LusternikSchnirelmann category ${\sf cat}_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semidirect product $\pi\rtimes G$ consisting of subconjugates of $G$. When $G$ is finite, we extend theorems of EilenbergGanea and StallingsSwan 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 StallingsSwan type result for families of subgroups which do not contain all finite subgroups.
Original language  English 

Journal  Groups, Geometry and Dynamics 
Publication status  Accepted/In press  23 Dec 2020 
Keywords
 55N91
 20J05 (Primary)
 55M30
 20E36 (Secondary)
 equivariant group cohomology
 equivariant Lusternik–Schnirelmann category
 classifying spaces
Fingerprint
Dive into the research topics of 'Equivariant dimensions of groups with operators'. Together they form a unique fingerprint.Profiles

Mark Grant
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
Person: Academic