The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions

2025/09/02 by Kunal Chawla, Behrang Forghani, Chawla, Kunal +5
Mathematics · #05C81 #22E40 #60G50 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2509.02500

openalex publication_date 2025/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that the Poisson boundary of random walks of finite entropy on Zariski-dense discrete subgroups of semisimple Lie groups equals the Furstenberg boundary of the corresponding symmetric spaces equipped with the hitting measure, without assuming any moment condition on the random walk.

Citations

• Non-Realizability of the Poisson Boundary
• Secret sharing on the Poisson-Furstenberg boundary I
• Limit theorems for a strongly irreducible product of independent random matrices under optimal moment assumptions
• The Poisson boundary of wreath products
• The Poisson boundary of hyperbolic groups without moment conditions
• Examples of measures with trivial left and non-trivial right random walk tail boundary
• Random walks on weakly hyperbolic groups
• The Poisson formula for groups with hyperbolic properties

Related

• The Poisson boundary of hyperbolic groups without moment conditions
• A note on the singularity conjecture for infinite covolume discrete subgroups
• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
• Quotients of Poisson boundaries, entropy, and spectral gap
• The Poisson formula for groups with hyperbolic properties