Mapping tori with first Betti number at least two

We show that given a finitely presented group G with 01 (G) ! 2 which is a mapping torus Gamma(theta) for Gamma a finitely generated group and theta an automorphism of Gamma then if the Alexander polynomial of G is non-constant, we can take beta(1)(Gamma) to be arbitrarily large. We give a range of applications and examples, such as any group G with beta(1)(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.