For every n≥ 2, every cocompact lattice Gamma<Isom( H^n), and every finitely supported admissible probability measure mu on Gamma, the hitting measure nu_mu of the associated random walk on partial H^n is singular with respect to Lebesgue measure lambda_partial H^n; that is, nu_mu perplambda_partial H^n.

Kaimanovich–Le Prince singularity conjecture

The conjecture predicts that the boundary hitting measure of broad classes of random walks on discrete groups is singular with respect to the natural boundary measure. Earlier work established important non-cocompact and special cocompact cases, while the general cocompact and lattice cases remained open.

Known results

Non-cocompact lattices, including Patterson–Sullivan measures for Fuchsian groups of the second kind, are essentially settled.
Kosenko obtained affirmative results for nearest-neighbour walks on cocompact Fuchsian groups.
Earlier work treated certain symmetric cocompact Fuchsian walks.
Open instances included selected regular tessellations, non-nearest-neighbour walks, and nonsymmetric fundamental polygons.

September 30, 2026: broad claimed extensions

Bogachev’s preprint claims singularity for cubulated, Kleinian, arithmetic, reflection, and related hyperbolic lattices using a von Neumann dimension argument extending Kosenko–Tiozzo. Other September preprints claim the unrestricted finite-support statement for discrete subgroups of $\mathrm{PSL}_{2}(\mathbb{R})$ and the cocompact Fuchsian case, implying all Fuchsian groups of the first kind. These are substantial claimed advances, not independently verified resolutions.

Current status (as of October 2026): Several major families, including the claimed Fuchsian cases and broad lattice classes, have recent unverified proofs; the full conjecture remains open outside those hypotheses.

Kaimanovich–Le Prince singularity conjecture

Primary source

Additional references

Progress summary

Solutions 0