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Some results related to finiteness properties of groups for families of subgroups. (English) Zbl 1523.20073

Summary: 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 D. Juan-Pineda and I. J. Leary [Contemp. Math. 407, 135–145 (2006; Zbl 1107.19001)] 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-\(\mathbb{Z}\)-group \(G\), that \(\underline{\underline{B}}G\) is homotopy equivalent to a finite CW-complex if and only if \(G\) is cyclic.

MSC:

20F65 Geometric group theory
20J05 Homological methods in group theory

Citations:

Zbl 1107.19001
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References:

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