On the finiteness of the classifying space for the family of virtually cyclic subgroups

Given a group G, we consider its classifying space for the family of virtually cyclic subgroups. We show for many groups, including for example, one-relator groups, acylindrically hyperbolic groups, 3-manifold groups and CAT(0) cube groups, that they do not admit a finite model for this classifying space unless they are virtually cyclic. This settles a conjecture due to Juan-Pineda and Leary for these classes of groups.Comment: Minor changes, to appear in Groups, Geometry, and Dynamic

On the Finiteness of the Classifying Space for Virtually Cyclic Subgroups

This thesis mainly deals with finiteness properties of the classifying space for the family of virtually cyclic subgroups. We establish a link between the finiteness of the classifying space and the conjugacy growth invariant of the given group and use this connection to show that the classifying space for virtually cyclic subgroups is not of finite type except in trivial cases if the group is linear or a CAT(0) cube group. We also investigate the class of residually finite groups in this context and along the way come close to a classification of the finite groups which have only two conjugacy classes of maximal cyclic subgroups

Some results related to finiteness properties of groups for families of subgroups

Von Puttkamer T, Wu X. Some results related to finiteness properties of groups for families of subgroups. Algebraic & Geometric Topology. 2020;20(6):2885-2904.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