A note on covers of fibred hyperbolic manifolds

Abstract

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, ${\phi }_1$ and ${\phi }_2$, and two non-equivalent covers $p_i:\tilde{S}⟶S$, $i=1,2$, so that the lift of ${\phi }_1$ to $\tilde{S}$ with respect to $p_1$ coincides with that of ${\phi }_2$ with respect to $p_2$. The mapping tori of the ${\phi }_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.

9 pages, 2 figures