Speaker: Irakli Patchkoria (University of Aberdeen)
Title: Equivariant Stallings-Swan theorem and cohomology of orbit categories
Abstract: Given a discrete group G and a family of subgroups F, a conjecture in geometric group theory asserts that the geometric dimension of G with respect to F is equal to 1 if and only if the cohomological dimension of the orbit category O_F(G) associated to F is equal to 1. The conjecture is known to be true for certain families: the trivial family (Stallings-Swan), the family of all finite subgroups (Dunwoody), and the family of virtually cyclic subgroups (Degrijse) among others. We prove the conjecture for any family F which does not contain the family of all finite subgroups. As a consequence one obtains a cohomological characterisation of free groups with an action of finite group and invariant basis. This is an equivariant version of the classical Stallings-Swan theorem. We will also discuss applications to the equivariant LS category. The talk will introduce basic concepts at the beginning and should be accessible to a general audience. This is all joint work with Mark Grant and Ehud Meir.