PUB - Publikationen an der Universität Bielefeld

Let (E)double-under-barG 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)double-under-barG if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of Luck, 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)double-under-barG if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space (B)double-under-barG = (E)double-under-barG/G. We show, for a poly-Z -group G, that (B)double-under-barG is homotopy equivalent to a finite CW-complex if and only if G is cyclic.