Gromov-Thuston manifolds have been introduced in the 80s as new examples of negatively curved Riemannian manifolds that are not homotopy equivalent to symmetric spaces. Topologically, they are constructed as branched coverings of a hyperbolic manifold along a suitable chosen codimension 2 totally geodesic submanifold. In a joint work with M. Incerti-Medici we compute the entropy of general branched coverings with curvature bounded above. I will also discuss an ongoing project where we try to understand if two different branched coverings on the base couple have quasi-isometric fundamental groups: this is already a relevant open question in the case of Gromov-Thurston manifolds.
