Scalar wave falloff in topological black hole backgrounds

We investigate the behavior of conformal scalar waves in topological black hole spacetimes. These spacetimes are all asymptotically anti–de Sitter spacetimes, and have black hole event horizons which are compact two-surfaces of arbitrary genus. We solve numerically the conformal scalar wave equation and show that its late time behavior is the same near or far from the event horizon. This behavior is strongly dependent upon the genus $g$ of the black hole, with higher genus black holes exhibiting a less rapid exponential falloff rate with a lower frequency of oscillation. In general, waves respecting Dirichlet boundary conditions decay more rapidly than those obeying Neumann conditions. We consider also negative mass black holes (allowed solutions only for higher genus), and find that the behavior of the wave is considerably different from the positive mass cases, undergoing a transition from falloff to amplification.