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

In this talk, I will first give a gentle introduction to  the classifying space for the family of subgroups with many
examples. Note that when the family consists of only the trivial subgroup, the classifying space is just the
Eilenberg-Maclane space which is a classical object to study. Then I will concentrate on the case when  the
family is the set of  virtually cyclic subgroups. In particular, I will discuss an interesting conjecture due to
Juan-Pineda and Leary. The conjecture says a group admits a finite model for the classifying space for the
family of virtually cyclic subgroups if and only if it is virtually cyclic.  I will talk about some recent progress
on this conjecture. This is a joint work with Tim von Puttkamer.