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

38th Annual Geometric Topology Workshop (Online), June 15-17, 2021

Ignat Soroko, Louisiana State University
Title: Groups of type FP: their quasi-isometry classes and homological Dehn functions
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 $FP_2$. 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 a joint paper with N.Brady, R.Kropholler and myself, we show that for any even integer $k\geq 4$ there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function $n^k$. In particular there exists an FP group with the quartic homological Dehn function and the unsolvable word problem. 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.