Topological Black Holes of Einstein-Yang-Mills dilaton Gravity

We present the topological solutions of Einstein-dilaton gravity in the presence of a non-Abelian Yang-Mills field. In 4 dimensions, we consider the $So(3)$ and $So(2,1)$ semisimple group as the Yang-Mills gauge group, and introduce the black hole solutions with spherical and hyperbolic horizons, respectively. The solution in the absence of dilaton potential is asymptotically flat and exists only with spherical horizon. Contrary to the non-extreme Reissner-Nordstrom black hole, which has two horizons with a timelike and avoidable singularity, here the solution may present a black hole with a null and unavoidable singularity with only one horizon. In the presence of dilaton potential, the asymptotic behavior of the solutions is neither flat nor anti-de Sitter. These solutions contain a null and avoidable singularity, and may present a black hole with two horizons, an extreme black hole or a naked singularity. We also calculate the mass of the solutions through the use of a modified version of Brown and York formalism, and consider the first law of thermodynamics.