Linear Groups, Conjugacy Growth, and Classifying Spaces for Families of Subgroups

TL;DR

This paper explores the relationship between conjugacy growth and classifying spaces for families of subgroups, proving a conjecture for linear groups and constructing examples with specific automorphism properties.

Contribution

It establishes that the Juan-Pineda and Leary conjecture holds for linear groups by linking conjugacy growth to classifying spaces, and constructs finitely generated groups with wild automorphisms but finite models.

Findings

Juan-Pineda and Leary conjecture holds for linear groups.

Constructed finitely generated groups with wild automorphisms and finite classifying space models.

Connected conjugacy growth to the existence of finite models for classifying spaces.

Abstract

Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by L\"uck--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we construct finitely generated groups that exhibit wild inner automorphims but which admit a model for $E_{\mathcal{VC}yc}(G)$ whose 0-skeleton is finite.