Chabauty limits of algebraic groups acting on trees

TL;DR

This paper investigates the limits of algebraic groups acting on trees within the Chabauty topology, revealing how certain algebraic groups can be approximated and how their structural properties change in different residue characteristics.

Contribution

It establishes that quasi-split simple algebraic groups form a closed set in the Chabauty space of automorphisms of trees and analyzes the behavior of Tits indices under limits in residue characteristic 2.

Findings

Quasi-split simple algebraic groups form a closed subset in the Chabauty space.

The Tits index of algebraic subgroups may not be preserved in residue characteristic 2.

The study extends Bruhat–Tits models over arbitrary local fields without restrictions.

Abstract

Given a locally finite leafless tree $ T $, various algebraic groups over local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $ \text{Aut} (T)$. This is done via a study of the integral Bruhat$-$Tits model of $ \text{SL}_2 $ and $ \text{SU}_3^{L/K} $, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $ 2 $, the Tits index of simple algebraic subgroups of $ \text{Aut} (T)$ is not always preserved under Chabauty limits.