#### Volume 20, issue 6 (2020)

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

### Timm von Puttkamer and Xiaolei Wu

Algebraic & Geometric Topology 20 (2020) 2885–2904
DOI: 10.2140/agt.2020.20.2885
##### Abstract

Let $\underset{¯}{\underset{¯}{E}}G$ be the classifying space of $G$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $\underset{¯}{\underset{¯}{E}}G$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Lück, Reich, Rognes and Varisco for Artin groups. We then study 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 $\underset{¯}{\underset{¯}{E}}G$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $\underset{¯}{\underset{¯}{B}}G=\underset{¯}{\underset{¯}{E}}G∕G\phantom{\rule{-0.17em}{0ex}}$. We show, for a poly-$ℤ$–group $G\phantom{\rule{-0.17em}{0ex}}$, that $\underset{¯}{\underset{¯}{B}}G$ is homotopy equivalent to a finite CW–complex if and only if $G$ is cyclic.

##### Keywords
finiteness properties of groups for families of subgroups, Artin groups, conjugacy growth, CAT(0) cube group, virtually cyclic groups, poly-$\mathbb{Z}$–groups
##### Mathematical Subject Classification 2010
Primary: 20B07, 20J05