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

Timm von Puttkamer and Xiaolei Wu

Algebraic & Geometric Topology 20 (2020) 2885–2904
DOI: 10.2140/agt.2020.20.2885

Let 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 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 E¯ ¯G if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space B¯ ¯G = E¯¯GG. We show, for a poly-–group G, that B¯ ¯G is homotopy equivalent to a finite CW–complex if and only if G is cyclic.

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
Received: 19 October 2018
Revised: 20 April 2019
Accepted: 1 December 2019
Published: 8 December 2020
Timm von Puttkamer
Mathematical Institute
University of Bonn
Xiaolei Wu
Mathematical Institute
University of Bonn
Faculty of Mathematics
Bielefeld University