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

We introduce the notion of the k-closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the k-closure satisfies a new property of automorphism groups of trees that generalises Tits' Property P. We prove that, apart from some degenerate cases, any non-discrete group acting on a tree with this property contains an abstractly simple subgroup.