Detecting Stationary Measure Classes

A random walk on a hyperbolic group G will converge almost surely to the boundary, defining a stationary measure at infinity. When G is a cocompact fuchsian group, this boundary is the circle, and when the random walk is finitely supported it is unknown whether this measure can ever be absolutely continuous with respect to the Lebesgue measure. We provide a simple and geometric family of statistics which determine whether two stationary measures are in the same measure class.