Mapping tori with first Betti number at least two

Abstract

We show that given a finitely presented group $G$ with ${\beta }_1\left(G\right)\geqq 2$ which is a mapping torus ${\mathrm{\Gamma }}_{\theta }$ for $\mathrm{\Gamma }$ a finitely generated group and $\theta$ an automorphism of $\mathrm{\Gamma }$ then if the Alexander polynomial of $G$ is non-constant, we can take ${\beta }_1\left(\mathrm{\Gamma }\right)$ to be arbitrarily large. We give a range of applications and examples, such as any group $G$ with ${\beta }_1\left(G\right)\ge 2$ that is $F_n$-by-$\mathbf{Z}$ for $F_n$ the non-abelian free group of rank $n$ is also $F_m$-by-$\mathbf{Z}$ 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.