Kaimanovich–Le Prince singularity conjecture for hitting measures

Let $\Gamma<\mathrm{SL}_N(\mathbb{R})$ be a discrete subgroup, and let the random walk on $\Gamma$ have finite support. Its hitting measure is a measure on the boundary at infinity, equipped with Lebesgue measure. Kaimanovich–Le Prince conjecture. The hitting measure of any finitely supported random walk on a discrete subgroup $\Gamma<\mathrm{SL}_N(\mathbb{R})$ is singular at infinity with respect to the Lebesgue measure. This conjecture concerns the relationship between random-walk hitting measures and Lebesgue measure on the boundary at infinity; the paper proves the singularity conjecture for certain measures on “most” cocompact Fuchsian and Kleinian groups, while the general formulation remains open.