Shear viscosity and Chern-Simons diffusion rate from hyperbolic horizons

We calculate the shear viscosity and anomalous baryon number violation rate in quantum field theories at finite temperature having gravity duals with hyperbolic horizons. We find the explicit dependence of these quantities on the temperature. We show that the ratio of shear viscosity to entropy density is below 1/(4π) at all temperatures and can be made arbitrarily small in the low temperature limit for hyperbolic surfaces of sufficiently high genus so that the hydrodynamic limit remains valid.