A note on covers of fibred hyperbolic manifolds

TL;DR

This paper constructs pairs of conjugate pseudo-Anosov maps on surfaces of genus greater than 2, with non-equivalent covers, leading to examples of hyperbolic 3-manifolds with multiple covering relationships.

Contribution

It introduces a novel method to produce hyperbolic 3-manifolds with multiple distinct covers via conjugate pseudo-Anosov maps and their lifts.

Findings

Existence of pairs of conjugate pseudo-Anosov maps with specific cover properties

Construction of hyperbolic 3-manifolds with multiple covering structures

Examples of non-equivalent covers of mapping tori

Abstract

For each surface $S$ of genus $g>2$ we construct pairs of conjugate pseudo-Anosov maps, $\varphi_1$ and $\varphi_2$, and two non-equivalent covers $p_i: \tilde S \longrightarrow S$, $i=1,2$, so that the lift of $\varphi_1$ to $\tilde S$ with respect to $p_1$ coincides with that of $\varphi_2$ with respect to $p_2$. The mapping tori of the $\varphi_i$ and their lift provide examples of pairs of hyperbolic $3$-manifolds so that the first is covered by the second in two different ways.