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Poisson boundaries of building lattices and rigidity with hyperbolic-like targets

Antoine Derimay

TL;DR

The work identifies the Poisson-Furstenberg boundary for regular thick Euclidean buildings and their lattices as the chamber-at-infinity space with harmonic measures, and uses this boundary description to prove strong rigidity phenomena. Central to the approach are prouniform boundary measures and a boundary-geometry framework that replaces algebraic group structure with building-theoretic Weyl-group data. The main contributions are a detailed boundary-disintegration analysis, a proof that the Poisson boundary of a lattice matches the boundary of the building, and superrigidity results: any morphism or cocycle from a higher-rank building lattice into geometrically rigid groups virtually fixes a boundary point or has finite image. These results extend rigidity phenomena to non-linear, negatively curved targets and highlight the boundary as a unifying tool in higher-rank rigidity for building lattices.

Abstract

We prove that a Poisson boundary of any regular thick Euclidean building, as well as lattices thereof is the space of chambers at infinity of the building with the harmonic measure class. We then use this result to generalize rigidity results of Guirardel-Horbez-Lécureux on morphisms and cocycles from lattices in buildings to groups with negative curvature properties.

Poisson boundaries of building lattices and rigidity with hyperbolic-like targets

TL;DR

The work identifies the Poisson-Furstenberg boundary for regular thick Euclidean buildings and their lattices as the chamber-at-infinity space with harmonic measures, and uses this boundary description to prove strong rigidity phenomena. Central to the approach are prouniform boundary measures and a boundary-geometry framework that replaces algebraic group structure with building-theoretic Weyl-group data. The main contributions are a detailed boundary-disintegration analysis, a proof that the Poisson boundary of a lattice matches the boundary of the building, and superrigidity results: any morphism or cocycle from a higher-rank building lattice into geometrically rigid groups virtually fixes a boundary point or has finite image. These results extend rigidity phenomena to non-linear, negatively curved targets and highlight the boundary as a unifying tool in higher-rank rigidity for building lattices.

Abstract

We prove that a Poisson boundary of any regular thick Euclidean building, as well as lattices thereof is the space of chambers at infinity of the building with the harmonic measure class. We then use this result to generalize rigidity results of Guirardel-Horbez-Lécureux on morphisms and cocycles from lattices in buildings to groups with negative curvature properties.
Paper Structure (25 sections, 53 theorems, 148 equations)

This paper contains 25 sections, 53 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Buildings and measures
  3. Generalities about buildings
  4. Coxeter groups and complexes
  5. Root systems and Euclidean reflection groups
  6. Buildings
  7. The building at infinity
  8. Harmonic measures at infinity
  9. Prouniform measures
  10. Random walks
  11. Graphs and Markov chains
  12. Poisson-Furstenberg boundaries
  13. The induced walk on a recurrent subset
  14. A recurrence criterion
  15. The Poisson-Furstenberg boundary of a lattice
  16. ...and 10 more sections

Key Result

Theorem 1

Let $(X,P)$ be an irreducible random walk on a countable discrete space, and $\Gamma<\mathop{\mathrm{Aut}}\nolimits(X,P)$ be a discrete subgroup. If $\Gamma$ is a lattice in $X$, there is an admissible measure $\mu$ on $\Gamma$ such that where the isomorphism is $\Gamma$-equivariant. Moreover, $\mu$ is symmetric whenever $P$ is, and if $\Gamma$ is uniform then $\mu$ has a finite exponential momen

Theorems & Definitions (115)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Remark 1.1
  • Theorem 4
  • Remark 1.2
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • ...and 105 more