Let Gamma<PSL_2(R) be a cocompact Fuchsian group, and let mu be a finitely supported non-degenerate probability measure on Gamma. If nu_mu denotes the harmonic measure of the random walk driven by mu on the boundary partialH^2 cong S^1, and lambda denotes Lebesgue measure on S^1, then nu_mu and lambda are mutually singular: nu_mu perplambda.

Singularity conjecture for harmonic measure

The Guivarc’h–Kaimanovich–Ledrappier conjecture concerns singularity between harmonic measure from a random walk and Lebesgue measure on the boundary of a hyperbolic group. The new claim addresses the cocompact Fuchsian case, rather than every setting covered by the broader conjectural picture.

Known results

Randecker and Tiozzo, 2021: for finite-volume hyperbolic $N$ -manifolds, harmonic and Lebesgue measures are mutually singular for every $N \ge 2$ .
A 2020 theorem proves singularity, with Hausdorff dimension less than $1$ , for walks generated by side-pairing translations of suitable centrally symmetric hyperbolic polygons.
Results through 2024 covered special walks and families of groups, while describing the general cocompact Fuchsian case as open.

September 2026 preprint

Petr Kosenko and Giulio Tiozzo claim that their paper constructs distinct geodesic currents for Lebesgue and random-walk measures and uses boundary Fourier analysis to prove singularity for cocompact Fuchsian groups. This is presented as a resolution of that case, but remains an unverified preprint.

Current status (as of September 2026): The cocompact Fuchsian case is claimed solved by an unrefereed preprint, pending verification; broader versions of the conjecture remain open.

Singularity conjecture for harmonic measure

Primary source

Additional references

Progress summary

Solutions 0