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

For a group $G$ we consider the classifying space $E_{\mathcal{V}\mathcal{C}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{\mathcal{V}\mathcal{C}yc}(G)$ 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 $E_{\mathcal{V}\mathcal{C}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{\mathcal{V}\mathcal{C}yc}(G)=E_{\mathcal{V}\mathcal{C}yc}(G)\mathrm{/}G$. We show for a poly-$\mathbb Z$-group$G$,that$B_{$\mathcal{V}\mathcal{C}$yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.