Large Groups of Deficiency One

TL;DR

This paper characterizes groups with deficiency one presentations involving a commutator, showing conditions under which they are large or far from residually finite, with applications to various classes of groups.

Contribution

It establishes new criteria for largeness and residual finiteness in groups with deficiency one presentations involving commutators.

Findings

Groups with deficiency one and a commutator relator are either infinite cyclic, large, or far from residually finite.

Mapping tori of free group endomorphisms are large if they contain certain subgroups of infinite index.

Applications include results on free-by-cyclic groups, 1-relator groups, and residually finite groups.

Abstract

We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, is large, or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains the integers times the integers as a subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups.