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.
"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