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We introduce the notion of the $k$-closure of a group of automorphisms of alocally finite tree, and give several examples of the construction. We showthat the $k$-closure satisfies a new property of automorphism groups of treesthat generalises Tits' Property $P$. We prove that, apart from some degeneratecases, any non-discrete group acting on a tree with this property contains anabstractly simple subgroup.

Simple groups of automorphisms of trees determined by their actions on finite subtrees