Residual Deficiency

TL;DR

This paper introduces residual deficiency, a new invariant for finitely presented groups, which helps identify groups with rich subgroup structures and positive properties like largeness and positive Betti numbers.

Contribution

It defines residual deficiency, proves its key properties, computes it for known groups, and explores its behavior under subgroups and quotients.

Findings

Residual deficiency greater than one implies the group is large.

Residual deficiency minus one is supermultiplicative for finite index normal subgroups.

Provides lower bounds for residual deficiency of quotients.

Abstract

We introduce a new real valued invariant for finitely presented groups called residual deficiency. Its main property is the following. Let G be a finitely presented group. If the residual deficiency of G is greater than one, then G has a finite index subgroup with deficiency greater than one. The latter property is strong. For instance, such a group is large, has positive rank gradient and positive first L^2-Betti number, among other properties. We also compute the residual deficiency of some well known families of presentations, prove that residual deficiency minus one is supermultiplicative with respect to finite index normal subgroups and find lower bounds for the residual deficiency of quotients.