Largeness of LERF and 1-relator groups

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-$Z$ word hyperbolic groups which are large, show that a LERF deficiency 1 group with first Betti number at least two is large or $Z\times Z$ and show that 2-generator 1relator groups where the relator has height $1$ obey the dichotomy that either the group is large or all its finite images are metacyclic.