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Some results related to
finiteness properties of groups for families of subgroups
Timm von Puttkamer and Xiaolei Wu |
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Algebraic & Geometric Topology 20 (2020) 2885–2904
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DOI: 10.2140/agt.2020.20.2885
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Abstract |
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Let be the classifying space of for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for 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 if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space . We show, for a poly-–group , that is homotopy equivalent to a finite CW–complex if and only if is cyclic. |
Keywords
finiteness properties of groups for families of subgroups,
Artin groups, conjugacy growth, CAT(0) cube group,
virtually cyclic groups, poly-$\mathbb{Z}$–groups
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Mathematical Subject Classification 2010
Primary: 20B07, 20J05
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References |
Publication
Received: 19 October 2018
Revised: 20 April 2019
Accepted: 1 December 2019
Published: 8 December 2020
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Authors |
| Timm von Puttkamer | |
| Mathematical Institute University of Bonn Bonn Germany |
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| Xiaolei Wu | |
| Mathematical Institute University of Bonn Bonn Germany |
Faculty of Mathematics Bielefeld University Bielefeld Germany |
