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

von Puttkamer, Timm; Wu, Xiaolei

doi:10.2140/agt.2020.20.2885

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Timm von Puttkamer and Xiaolei Wu

Algebraic & Geometric Topology 20 (2020) 2885–2904

DOI: 10.2140/agt.2020.20.2885

Abstract

Let $\underline{\underline{E}} G$ be the classifying space of $G$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $\underline{\underline{E}} G$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Lück, Reich, Rognes and Varisco for Artin groups. We then study 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 $\underline{\underline{E}} G$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $\underline{\underline{B}} G = \underline{\underline{E}} G ∕ G$ . We show, for a poly- $ℤ$ –group $G$ , that $\underline{\underline{B}} G$ is homotopy equivalent to a finite CW–complex if and only if $G$ is cyclic.

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Keywords

finiteness properties of groups for families of subgroups, Artin groups, conjugacy growth, CAT(0) cube group, virtually cyclic groups, poly-$\mathbb{Z}$–groups

Mathematical Subject Classification 2010

Primary: 20B07, 20J05

References

Publication

Received: 19 October 2018

Revised: 20 April 2019

Accepted: 1 December 2019

Published: 8 December 2020

Authors

Timm von Puttkamer
Mathematical Institute University of Bonn Bonn Germany

Xiaolei Wu
Mathematical Institute University of Bonn Bonn Germany	Faculty of Mathematics Bielefeld University Bielefeld Germany

Volume 20, issue 6 (2020)