I have been looking at the numerical behavior of a particular quantity (of no direct importance here, though if you must know the gory details start with figure 17 here) associated to the geodesic flow on a surface of constant negative curvature and genus $g$. The behavior is quantitatively similar for $g = 2,3,4$ and physical intuition based on this quantity suggests that some "intrinsic" timescale--prime candidates are the mixing or relaxation time--should therefore depend on $g$ only weakly or even not at all.

So: what is known about the behavior of the mixing and relaxation (or similar) times associated to these flows as $g$ varies?