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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.