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

TL;DR

The paper proves that for a discrete, Zariski-dense subgroup $\Gamma$ of a semisimple Lie group $G$, any probability measure $\mu$ with finite entropy and generating $\Gamma$ as a semigroup yields the Furstenberg boundary $(G/P, u)$ as the Poisson boundary of $(\Gamma,\mu)$, without any moment conditions. It introduces a novel pin-down partition strategy that, instead of pivot-based arguments, uses critical times and flats to show the conditional entropy along the boundary is sublinear, thereby obtaining $h_X=0$ and maximality of the boundary. The technique relies on the Cartan projection and projection to flats, along with bilateral random walks, to control asymptotics in higher rank, and it extends classical boundary identifications to settings with only finite entropy. The result generalizes Poisson boundary identifications in rank one and acylindrically hyperbolic groups to higher rank semisimple groups, offering a robust probabilistic-geometric framework for boundary analysis with no moment assumptions. The work has implications for rigidity phenomena and the probabilistic structure of random walks on lattices and discrete subgroups of Lie groups.

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.

Theorems & Definitions (10)

• Theorem 1.1
• proof
• Theorem 3.3
• Theorem 4.2
• proof
• proof
• proof
• proof
• Theorem 5.8
• proof : Proof of Theorem \ref{['T:main']}