Mapping tori with first Betti number at least two

Abstract

We show that given a finitely presented group G with &\beta;₁(G)≥2 which is a mapping torus Γ_θ for Γ a finitely generated group and θ an automorphism of Γ then if the Alexander polynomial of G is non-constant, we can take β₁(Γ) to be arbitrarily large. We give a range of applications and examples, such as any group G with β₁(G)≥2 that is F_n-by-Z for F_n the non-abelian free group of rank n is also F_m-by-Z for infinitely many m. We also examine 3-manifold groups where we show that a finitely generated subgroup cannot be conjugate to a proper subgroup of itself.