Virtual Betti numbers of genus 2 bundles

Abstract

We show that if $M$ is a surface bundle over $S^1$ with fiber of genus 2, then for any integer $n$, $M$ has a finite cover $\tilde{M}$ with $b_1\left(\tilde{M}\right)>n$. A corollary is that $M$ can be geometrized using only the “non-fiber" case of Thurston’s Geometrization Theorem for Haken manifolds.