Groups of type FP: their quasi-isometry classes and homological Dehn functions

Ignat Soroko (Louisiana State)


There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type \(F_2\). Considering a homological analog of finite presentability we get the class of groups \(FP_2\). Ian Leary proved that there are uncountably many isomorphism classes of groups of type \(FP_2\) (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In an on-going project with N.Brady, R.Kropholler and myself, we show that for any integer \(k\ge4\) there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function \(n^k\). In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem.