I was asked by a friend this question: Suppose G is a subgroup of $SL_2(\mathbb{R})$ acting on the upper half-plane, such that the boundary of the fundamental domain of the action has zero measure. Is $L^2$(G\ $\mathbb{H}$) a separable Hilbert space?
I am thinking the space is separable but am not sure. My idea is to show it's isomorphic to some separable space, but I don't have a clear picture. Any hints or suggestions on this? Thanks in advance.