Largeness of LERF and 1-relator groups

Abstract

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 Z 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. Mathematics Subject Classification (2010). 20F05.

On surface subgroups of doubles of free groups

“…. As pointed out in [5], the argument of [14, Proposition 2.1 and Theorem 2.2] then shows that G 2 (w) is large.…”

“…More generally, one can ask how common it is for a two-generator one-relator group to be large or to have virtual first Betti number greater than one. Button has used the Alexander polynomial to study large one-relator groups [5,6]. In particular, he has shown that the vast majority of two-generator one-relator presentations in 'Magnus form' with cyclically reduced relation of length at most 12 are large (see [6,Theorem 3.3]).…”

The Tits alternative for non-spherical triangles of groups

“…Definition 3.11 ('angle and swap'). By (10) and (11) j} ( a s )). Now, we can use the injectivity of the homomorphism ν {i,j} : G {i,j} → G to conclude that a s = ϕ {i}{i,j} ( a s ).…”

“…Definition 3.11 ('angle and swap'). By (10) and (11), we know that every local vertex D k is of some type {i, j} with distinct i, j ∈ I. By (9), such a local vertex is incident with bridges each of which is either of type i or of type j.…”

Tubular groups, 1-relator groups and nonpositive curvature

“…Thus on regarding "well behaved geometrically" to mean CAT(0) and "well behaved group theoretically" to mean residually finite, we have 1-relator groups R p,p which are well behaved group theoretically but not geometrically, 1-relator groups R p,3p which are well behaved geometrically but not group theoretically, and (for p > 3) 1relator groups R p,p−1 which are neither, all without unbalanced Baumslag -Solitar subgroups. This construction also allows us to answer Question 20 in [7], namely (for p even) the 2-generator 1-relator group R p,p/2 is residually finite but has no finite index subgroup which is an ascending HNN extension of a finitely generated free group.…”