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Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries

Corina Ciobotaru, Corentin Le Bars

TL;DR

This work addresses the asymptotic and operator-algebraic structure of groups acting on affine buildings by introducing and exploiting the notion of general type and the flag limit set. The authors develop a robust boundary framework built from strongly regular hyperbolic dynamics and an equivariant barycenter map on generic triples, which yields C*-simplicity, mean proximality, and explicit Poisson boundary descriptions for cocompact lattices. The main technical engine combines boundary theory (Furstenberg, stationary measures) with CAT(0) geometry and apartment-based retractions to produce universal proximal sequences and equicontinuous decompositions. These results generalize classical lattice cases to exotic, non-lattice actions, providing new insights into rigidity, operator algebras, and random walks on groups acting on buildings. The approach unifies dynamical, geometric, and probabilistic methods to illuminate the boundary dynamics and compute Poisson boundaries in a broad, structurally rich setting.

Abstract

We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of $X$. For a group $G$ of general type, we prove C*-simplicity by showing that its flag limit set $Λ_{\mathcal F}(G)$ is topologically free, minimal, and strongly proximal. When $Λ_{\mathcal F}(G)$ intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building.

Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries

TL;DR

This work addresses the asymptotic and operator-algebraic structure of groups acting on affine buildings by introducing and exploiting the notion of general type and the flag limit set. The authors develop a robust boundary framework built from strongly regular hyperbolic dynamics and an equivariant barycenter map on generic triples, which yields C*-simplicity, mean proximality, and explicit Poisson boundary descriptions for cocompact lattices. The main technical engine combines boundary theory (Furstenberg, stationary measures) with CAT(0) geometry and apartment-based retractions to produce universal proximal sequences and equicontinuous decompositions. These results generalize classical lattice cases to exotic, non-lattice actions, providing new insights into rigidity, operator algebras, and random walks on groups acting on buildings. The approach unifies dynamical, geometric, and probabilistic methods to illuminate the boundary dynamics and compute Poisson boundaries in a broad, structurally rich setting.

Abstract

We investigate a class of groups acting on possibly exotic affine buildings and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of . For a group of general type, we prove C*-simplicity by showing that its flag limit set is topologically free, minimal, and strongly proximal. When intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building.
Paper Structure (32 sections, 48 theorems, 57 equations, 2 figures)

This paper contains 32 sections, 48 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Groups of general type and their flag limit sets
  3. $C^\ast$-simplicity for groups acting on buildings
  4. Mean proximality
  5. Poisson boundaries of lattices
  6. Further developments
  7. Affine buildings
  8. Spherical building at infinity
  9. The cone topology on $\overline{X}$.
  10. Retractions
  11. Regularity
  12. Strongly regular hyperbolic elements
  13. Residue building based at a vertex
  14. Harmonic measures on $\text{ch}(X^\infty)$
  15. Topological and measurable boundaries
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $X$ be a locally finite affine building and let $G < \mathop{\mathrm{Isom}}\nolimits(X)$ be a group of general type. Then the limit set $\Lambda_\mathcal{F}(G)$ is the smallest closed non-empty $G$-invariant subset of $\text{ch}(X^\infty)$. It is $G$-minimal, perfect and uncountable. If $\Lambda

Figures (2)

  • Figure 1: The alcove $c_y$ and a subset of $\overline{\text{Opp}_o(c_y)}$ in yellow.
  • Figure 2: The apartment corresponding to $T$ in the spherical building of $G$

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem 1.2: $C^\ast$-simplicity
  • Theorem 1.3
  • Corollary 1.4
  • proof
  • Proposition 1.5
  • Proposition 1.6: Barycenter map
  • Proposition 1.7
  • Theorem 1.8: Mean proximality
  • Theorem 1.9: Poisson boundaries of lattices
  • ...and 84 more