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

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Puttkamer,  Timm von
Max Planck Institute for Mathematics, Max Planck Society;

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Wu,  Xiaolei
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1807.10095.pdf
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Citation

Puttkamer, T. v., & Wu, X. (2020). Some results related to finiteness properties of groups for families of subgroups. Algebraic & Geometric Topology, 20(6), 2885-2904. doi:10.2140/agt.2020.20.2885.

Cite as: http://hdl.handle.net/21.11116/0000-0007-A381-E
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.