Some results related to finiteness properties of groups for families of subgroups
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 JuanPineda and Leary and a question of LückReichRognesVarisco 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 CWcomplex if and only if $G$ is cyclic.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.10095
 Bibcode:
 2018arXiv180710095V
 Keywords:

 Mathematics  Group Theory;
 55R35;
 20B07
 EPrint:
 20 pages