Chabauty limits of simple groups acting on trees

Let \(T\) be a locally finite tree without vertices of degree 1. We show that among the closed subgroups of 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 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.