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

Ignat Soroko (Louisiana State)

Abstract

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 $F{P}_2$. Ian Leary proved that there are uncountably many isomorphism classes of groups of type $F{P}_2$ (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 $k\ge 4$ 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.