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

DOI10.2140/AGT.2020.20.2885arXiv1807.10095MaRDI QIDQ6304686

Timm von Puttkamer, Xiaolei Wu

Publication date: 26 July 2018

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

Mathematics Subject Classification ID

Geometric group theory (20F65) Homological methods in group theory (20J05)

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