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**A note on covers of fibred hyperbolic manifolds.**
*(English)*
Zbl 1380.57017

The main result of the paper states that, for each closed orientable surface \(S\) of genus at least three, there exist conjugate pseudo-Anosov diffeomorphisms \(\phi_1\) and \(\phi_2\) of \(S\) which lift, with respect to non-equivalent regular coverings \(p_1, p_2:\tilde S \to S\) (distinguished by non-isomorphic covering groups), to the same diffeomorphism \(\phi\) of the closed surface \(\tilde S\) (in fact, there are infinitely many pairs of such diffeomorphisms which are obtained by lifting Anosov diffeomorphisms of a torus to its branched covers). The mapping tori of the diffeomorphisms \(\phi_1\), \(\phi_2\) and their common lift \(\phi\) provide then pairs of fibered hyperbolic 3-manifolds such that the first is a regular covering of the second in two inequivalent ways (again with non-isomorphic covering groups; such examples were known previously for the case of non-fibered hyperbolic 3-manifolds).

Reviewer: Bruno Zimmermann (Trieste)

### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57M10 | Covering spaces and low-dimensional topology |