Some results related to finiteness properties of groups for families of subgroups

TL;DR

This paper investigates finiteness properties of classifying spaces for virtually cyclic subgroups across various groups, proving new results for Artin groups, CAT(0) groups, and poly-Z groups, with implications for their homotopy types and conjugacy growth.

Contribution

It establishes that only virtually cyclic Artin groups admit finite models for their classifying space, analyzes conjugacy growth in CAT(0) groups, and characterizes the homotopy type of quotient spaces for poly-Z groups.

Findings

Artin groups admit finite models for classifying spaces only if virtually cyclic.

CAT(0) groups with rank-two free abelian subgroups have super-linear conjugacy growth.

Poly-Z groups have finite CW-complexes homotopy equivalent to their classifying space only if cyclic.

Abstract

For a group $G$ we consider the classifying space $E_{\mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of L\"uck-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for $E_{\mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{\mathcal{VC}yc}(G) = E_{\mathcal{VC}yc}(G)/G$. We show for a poly-$\mathbb Z$-group $G$, that $B_{\mathcal{VC}yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.