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Quotients of Poisson boundaries, entropy, and spectral gap

Samuel Dodds, Alex Furman

Abstract

Poisson boundary is a measurable $Γ$-space canonically associated with a group $Γ$ and a probability measure $μ$ on it. The collection of all measurable $Γ$-equivariant quotients, known as $μ$-boundaries, of the Poisson boundary forms a partially ordered set, equipped with a strictly monotonic non-negative function, known as Furstenberg or differential entropy. In this paper we demonstrate the richness and the complexity of this lattice of quotients for the case of free groups and surface groups and rather general measures. In particular, we show that there are continuum many unrelated $μ$-boundaries at each, sufficiently low, entropy level, and there are continuum many distinct order-theoretic cubes of $μ$-boundaries. These $μ$-boundaries are constructed from dense linear representations $ρ:Γ\to G$ to semi-simple Lie groups, like $\PSL_2(\bbC)^d$ with absolutely continuous stationary measures on $\hat\bbC^d$.

Quotients of Poisson boundaries, entropy, and spectral gap

Abstract

Poisson boundary is a measurable -space canonically associated with a group and a probability measure on it. The collection of all measurable -equivariant quotients, known as -boundaries, of the Poisson boundary forms a partially ordered set, equipped with a strictly monotonic non-negative function, known as Furstenberg or differential entropy. In this paper we demonstrate the richness and the complexity of this lattice of quotients for the case of free groups and surface groups and rather general measures. In particular, we show that there are continuum many unrelated -boundaries at each, sufficiently low, entropy level, and there are continuum many distinct order-theoretic cubes of -boundaries. These -boundaries are constructed from dense linear representations to semi-simple Lie groups, like with absolutely continuous stationary measures on .

Paper Structure

This paper contains 23 sections, 29 theorems, 206 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Main results
  3. Our approach
  4. Notations and some Preliminary facts
  5. Stationary spaces, $\mu$-harmonic functions, and Poisson boundary
  6. The lattice of boundaries
  7. Entropy
  8. Integrability conditions on the measure $\mu$
  9. Semi-simple Lie groups
  10. Random walks
  11. Representations
  12. Spectral Gap
  13. Regularity of stationary measures
  14. Continuity of Entropy
  15. Estimating Entropy range
  16. ...and 8 more sections

Key Result

Theorem A

Let $\Gamma=F_n$ be a free group on $n\ge 3$ generators or a fundamental group $\Gamma=\pi_1(\Sigma_g)$ of a surface of genus $g\ge 2$, and $\mu$ a probability measure on $\Gamma$ that generates $\Gamma$ as a semi-group and has a finite exponential moment. Then there exists $h_1=h_1(\Gamma,\mu)>0$ a of distinct $\mu$-boundaries such that:

Theorems & Definitions (50)

  • Example 1.1
  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Corollary 2.4
  • Lemma 2.5
  • ...and 40 more