Gromov and Thurston famously used hyperbolic branched cover manifolds to construct the first examples of manifolds which admit a pinched negatively curved metric, but do not admit any locally symmetric metric. Much more recently, Fine and Premoselli (n=4) and Hamenstadt and Jackel (n>4) proved that many of these hyperbolic branched covers admit negatively curved Einstein metrics.

In this talk I will give an overview of these results and show how, in joint work with Lafont, we extended the construction of Fine and Premoselli to complex hyperbolic branched covers. This gives an explicit description of the first known negatively curved Kahler-Einstein metric on a manifold which does not admit a locally symmetric metric, whose existence was first proved by Guenancia and Hamenstadt.