Geodesic
Some results related to finiteness properties of groups for families of subgroups
von Puttkamer, Timm1 ; Wu, Xiaolei2
1Mathematical Institute, University of Bonn, Bonn, Germany
2Mathematical Institute, University of Bonn, Bonn, Germany, Faculty of Mathematics, Bielefeld University, Bielefeld, Germany
Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 2885-2904
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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¯¯G∕G. We show, for a poly-ℤ–group G, that B¯ ¯G is homotopy equivalent to a finite CW–complex if and only if G is cyclic.

Classification : 20B07, 20J05
Keywords: finiteness properties of groups for families of subgroups, Artin groups, conjugacy growth, CAT(0) cube group, virtually cyclic groups, poly-$\mathbb{Z}$–groups

von Puttkamer, Timm  1   ; Wu, Xiaolei  2

1 Mathematical Institute, University of Bonn, Bonn, Germany
2 Mathematical Institute, University of Bonn, Bonn, Germany, Faculty of Mathematics, Bielefeld University, Bielefeld, Germany 
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von Puttkamer, Timm; Wu, Xiaolei. Some results related to finiteness properties of groups for families of subgroups. Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 2885-2904. http://geodesic.mathdoc.fr/item/AGT_2020_20_6_a3/

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