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arXiv · 2411.04250

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

Corentin Le Bars·Jean Lécureux·Jeroen Schillewaert
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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.

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Corentin Le Bars, Jean Lécureux, Jeroen Schillewaert. 2024-11-06. A Tits alternative for $\mathbb{R}$-buildings of type $\tilde{A}_2$. https://arxiv.org/abs/2411.04250

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