Chabauty limits of simple groups acting on trees

TL;DR

This paper studies the limits of simple groups acting on trees and characterizes their Chabauty closures, with implications for automorphism groups of graphs and buildings, revealing topological properties of these groups.

Contribution

It provides a detailed description of the Chabauty closure of topologically simple groups acting on trees and explores their topological and algebraic properties, extending to automorphism groups of graphs and buildings.

Findings

Chabauty-closure of simple groups is characterized by groups without proper open finite index subgroups.

The set of simple groups acting doubly transitively on the boundary is compact and Hausdorff.

Results extend to automorphism groups of graphs and applications to buildings.

Abstract

Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\mathrm{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree $\geq 3$, then the set of isomorphism classes of topologically simple closed subgroups of $\mathrm{Aut}(T)$ acting doubly transitively on $\partial T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid in the context of automorphism groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.