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A Tits alternative for $\mathbb{R}$-buildings of type $\tilde{A}_2$

Corentin Le Bars, Jean Lécureux, Jeroen Schillewaert

Abstract

Let $G$ be a group with a non-elementary action on a (not necessarily discrete) $\tilde{A}_2$-buildings. We prove that, given a random walk on $G$, isometries in $G$ are strongly regular hyperbolic with high probability. As a consequence, we prove a Tits alternative for $G$, as well as a local-to-global fixed point result. We also prove that isometries of (not necessarily complete) $\mathbb{R}$-buildings are semi-simple.

A Tits alternative for $\mathbb{R}$-buildings of type $\tilde{A}_2$

Abstract

Let be a group with a non-elementary action on a (not necessarily discrete) -buildings. We prove that, given a random walk on , isometries in are strongly regular hyperbolic with high probability. As a consequence, we prove a Tits alternative for , as well as a local-to-global fixed point result. We also prove that isometries of (not necessarily complete) -buildings are semi-simple.

Paper Structure

This paper contains 20 sections, 26 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Affine buildings
  3. Non-discrete affine buildings
  4. Spherical building at infinity
  5. The cone topology on $\overline{X}$.
  6. Retractions
  7. Regularity
  8. Residue building based at a vertex
  9. Panel trees
  10. Reduction: separability
  11. Reduction: completeness
  12. Stationary measures
  13. Generalities
  14. Stationary measures on $\tilde{A}_2$-buildings
  15. Hyperbolic elements
  16. ...and 5 more sections

Key Result

Theorem A

A countable group with a non-elementary action on a building of type $\tilde{A}_2$ contains a non-abelian free subgroup.

Figures (2)

  • Figure 1: The set $\widetilde{\text{Opp}}_o(C)$ is contained in $\widetilde{\text{Opp}}_y(C)$
  • Figure 2: Construction of independent strongly regular hyperbolic isometries

Theorems & Definitions (52)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C: Fixed point
  • Theorem D
  • Remark 1.3
  • Theorem E
  • Proposition 2.1: rousseau23
  • Proposition 2.2
  • ...and 42 more