Besides being natural objects to study, branched covers of closed hyperbolic manifolds M along codimension 2 totally geodesic submanifolds B provide examples of interesting geometric and topological phenomena. For instance, Gromov and Thurston showed that most of them admit a Riemannian metric of pinched negative curvature but no locally symmetric one. As such natural geometric invariants of the covering like its volume are not necessarily topological invariants and key topological analogs such as the simplicial volume are not so simple to compute exactly. This often makes it difficult to answer simple questions such as: Can cyclic branched covers of M along B of different degrees be homotopy equivalent? In joint work with Alessandro Sisto we develop some criteria and tools to answer this question in terms of the arithmetic of the degrees.
