Abstract commensurators of groups acting on rooted trees. (Q1855280)

The author has previously constructed a family of finitely-presented, infinite, simple groups, each realized as the commutator subgroup of a group generated by a Higman-Thompson group and a group acting on a locally-finite, spherically-homogeneous tree. The current paper is motivated by the goal of obtaining a more algebraic description of these groups. In this goal, the author partially succeeds in that he proves the following Theorem. Grigorchuk's group \(G\) of intermediate growth has commensurator which is a finitely presented (infinite) simple group generated by \(G\) and the Higman-Thompson group \(G_{2,1}\). That is, the commensurator of \(G\) has essentially the structure exhibited by the author's earlier constructions. The first step in the proof is to show that the Higman-Thompson group \(G_{n,1}\) appears often in certain commensurators: Theorem. For each integer \(n\geq 2\), the commensurator of every infinite group which is commensurable with its own \(n\)-th direct power contains a subgroup isomorphic to a Higman-Thompson group. The second step identifies, under appropriate circumstances, the abstract commensurator of a group with a very concrete relative commensurator: Theorem. The abstract commensurator of a nearly-level-transitive, weakly-branch group \(G\) is isomorphic to the relative commensurator of \(G\) in the homeomorphism group of the boundary of the tree on which \(G\) acts.