## Virtually Haken surgeries on once-punctured torus bundles

### Abstract

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of most'' commensurability classes of punctured torus bundles. In particular, we define an integer-valued complexity function on monodromies $f$ (essentially the length of the LR-factorization of $f_*$ in $PSL_2(\mathbb{Z})$), and use a computer to show that if the monodromy of $M$ has complexity at most 5, then $M$ is finitely covered by an element of $\mathcal{C}$. If the monodromy has complexity at most 12, then, with at most 36 exceptions, $M$ is finitely covered by an element of $\mathcal{C}$. We also give a method for computing algebraic boundary slopes'' in certain finite covers of punctured torus bundles.Comment: Much expanded. Added applications of the method described in v. 2. Showed that many punctured torus bundles have the property that all but finitely many Dehn fillings are virtually Haken. For the precise sense of the word "many", see the revised abstrac

Topics: Mathematics - Geometric Topology
Year: 2006
OAI identifier: oai:arXiv.org:math/0506443