Rotating black holes with Nil or SL(2,$\,\mathbb{R}$) horizons

We construct rotating black holes in $N=2$, $D=5$ minimal and matter-coupled gauged supergravity, with horizons that are homogeneous but not isotropic. Such spaces belong to the eight Thurston model geometries, out of which we consider the cases Nil and SL$(2,\mathbb{R})$. In the former, we use the recipe of arXiv:hep-th/0304064 to directly rederive the solution that was obtained by Gutowski and Reall in arXiv:hep-th/0401042 as a scaling limit from a spherical black hole. With the same techniques, the first example of a black hole with SL$(2,\mathbb{R})$ horizon is constructed, which is rotating and one quarter BPS. The physical properties of this solution are discussed, and it is shown that in the near-horizon limit it boils down to the geometry of arXiv:hep-th/0401042, with a supersymmetry enhancement to one half. Dimensional reduction to $D=4$ gives a new solution with hyperbolic horizon to the t$^3$ model that carries both electric and magnetic charges. Moreover, we show how to get a nonextremal rotating Nil black hole by applying a certain scaling limit to Kerr-AdS$_5$ with two equal rotation parameters, which consists in zooming onto the north pole of the S$^2$ over which the S$^3$ is fibered, while boosting the horizon velocity effectively to the speed of light.