# Algebra/Topology seminar

Speaker: Nansen Petrosyan (University of Southampton)

Title: Groups with cocompact classifying spaces for proper actions and a question of K. S. Brown

Abstract: Many discrete groups we like have finite virtual cohomological dimension. By a well-known construction due to Serre, such groups admit finite dimensional classifying spaces for proper actions. One can think of this construction as a generalisation of a classical result of Eilenberg and Ganea that a group G with finite cohomological dimension cd(G) has a finite dimensional classifying space EG of dimension equal to cd(G) provided cd(G) is not 2 in which case it is equal to 3. The difference here is that if the group G has torsion then Serre's construction produces a space of dimension at least twice as big as the vcd(G). In 1977, Ken Brown asked given a group G with finite vcd(G) whether one could always construct a model for the classifying space for proper actions of dimension equal to the vcd(G) and under which conditions on the group there exists such a cocompact model. In 2003, Leary and Nucinkis found the first examples of groups that gave a negative answer to the first part of Brown's question. Yet their construction produced groups which could not have a cocompact model for the classifying spaces for proper actions. In this talk I will discuss how one can construct families of finite extensions of right-angled Coxeter groups that have a cocompact model for the classifying space for proper actions of minimal dimension but their virtual cohomological dimension is strictly less than this dimension. In fact, for these groups the gap between the two dimensions can be arbitrarily large. This is joint work with Ian Leary.