A note on the singularity conjecture for infinite covolume discrete subgroups

2025/08/07 by Dongryul M. Kim, Kim, Dongryul M., Andrew Zimmer +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2508.05756

openalex publication_date 2025/08/07 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a finite first moment, we again show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. For general semisimple Lie groups, we also obtain a sufficient condition for the singularity of the stationary measure and a general Patterson-Sullivan measure.

Citations

• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
• Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity
• Remarks on discrete subgroups with full limit sets in higher rank Lie groups
• Dichotomy and measures on limit sets of Anosov groups
• Some asymptotic properties of random walks on homogeneous spaces
• The fundamental inequality for cocompact Fuchsian groups
• Invariant measures for horospherical actions and Anosov groups
• Cusp excursion in hyperbolic manifolds and singularity of harmonic measure
• Entropy and drift for Gibbs measures on geometrically finite manifolds
• Random walks on weakly hyperbolic groups
• Matrix random products with singular harmonic measure
• On the question of ergodicity for minimal group actions on the circle
• The Poisson formula for groups with hyperbolic properties

Cited by

• Random walks on cocompact Fuchsian and Kleinian groups
• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank

Related

• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
• The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions
• Random walks on cocompact Fuchsian and Kleinian groups
• Absolutely continuous Furstenberg measures for finitely-supported random walks
• Some asymptotic properties of random walks on homogeneous spaces