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2020 Some results related to finiteness properties of groups for families of subgroups
Timm von Puttkamer, Xiaolei Wu
Algebr. Geom. Topol. 20(6): 2885-2904 (2020). DOI: 10.2140/agt.2020.20.2885

Abstract

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.

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Timm von Puttkamer. Xiaolei Wu. "Some results related to finiteness properties of groups for families of subgroups." Algebr. Geom. Topol. 20 (6) 2885 - 2904, 2020. https://doi.org/10.2140/agt.2020.20.2885

Information

Received: 19 October 2018; Revised: 20 April 2019; Accepted: 1 December 2019; Published: 2020
First available in Project Euclid: 16 December 2020

MathSciNet: MR4185930
Digital Object Identifier: 10.2140/agt.2020.20.2885

Subjects:
Primary: 20B07, 20J05

Rights: Copyright © 2020 Mathematical Sciences Publishers

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