Covers of Dehn Fillings on Once-Punctured Torus Bundles

Abstract

Let $M$ be a compact, orientable $3$-manifold that fibers over ${S^1}$ with fiber a once-punctured torus, ${T_0}$, and characteristic homeomorphism $h:{T_0} \to {T_0}$. We prove that for certain characteristic homeomorphisms, most Dehn fillings on $M$ yield manifolds with virturally ${\mathbf {Z}}$-representable fundamental groups.