I also prove that, if G is an infinte group which is commensurable with its on n-th direct power for some n>1, the abstract commensurator of G contains a Higman-Thompson group as a subgroup.
This plus some extra work enables me to determine the abstract commensurator C of ‘the’ Grigorchuk 2-group. In particular, it is shown that C is a finitely presented (infinite) simple group.
This is published in Geom. Dedicata 94 (2002), 45-61
Of course you can also ask me for a copy.