Singularity of harmonic measures for hyperbolic lattices
Abstract
Let Γ<Isom(ℍn), n≥2, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on Γ has hitting measure singular with respect to Lebesgue measure on ∂ℍn. The proof extends the method of Kosenko--Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.
Related
- Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
- Singularity of harmonic measure for finitely supported random walks
- Transversal Hölder Criteria and Dini--Zygmund Endpoint Regularity for Hyperbolic Harmonic Mappings
- Cubical residual complexes and profinite rigidity of real hyperbolic lattices
- Strongly log-concave measures and lattices
BibTeX
@misc{bogachev2026singularity,
title = {{Singularity of harmonic measures for hyperbolic lattices}},
author = {Nikolay Bogachev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40357},
eprint = {2609.40357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40357},
}
RIS
TY - UNPB
TI - Singularity of harmonic measures for hyperbolic lattices
AU - Nikolay Bogachev
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40357
UR - https://arxiv.org/abs/2609.40357
AB - Let Γ<Isom(ℍ^n), n≥2, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on Γ has hitting measure singular with respect to Lebesgue measure on ∂ℍ^n. The proof extends the method of Kosenko--Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.
ER -