Largeness of LERF and 1-relator groups

TL;DR

This paper investigates the largeness property of certain groups with deficiency 1, including free-by-cyclic, LERF, and 1-relator groups, providing new examples and conditions for largeness.

Contribution

It presents the first examples of hyperbolic free-by-cyclic groups that are large and establishes conditions under which LERF and 1-relator groups are large.

Findings

First hyperbolic free-by-cyclic groups shown to be large

LERF deficiency 1 groups with Betti number ≥ 2 are large

2-generator 1-relator groups with height 1 relators are either large or have all finite images metacyclic

Abstract

We consider largeness of groups given by a presentation of deficiency 1, where the group is respectively free-by-cyclic, LERF or 1-relator. We give the first examples of (finitely generated free)-by-(infinite cyclic) word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least 2 is large or the integers times the integers, and show that 2-generator 1-relator groups where the relator has height 1 obey the dichotomy that either the group is large or all its finite images are metacyclic.