Groups, Geometry, and Dynamics

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Volume 4, Issue 4, 2010, pp. 709–738
DOI: 10.4171/GGD/102

Published online: 2010-10-15

Largeness of LERF and 1-relator groups

Jack O. Button[1]

(1) University of Cambridge, UK

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 × 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.

Keywords: 1-relator groups, largeness, LERF

Button Jack: Largeness of LERF and 1-relator groups. Groups Geom. Dyn. 4 (2010), 709-738. doi: 10.4171/GGD/102