Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity

The Margulis-Zimmer conjecture. The subject of this paper is a well known question advertised by Gregory Margulis and Robert Zimmer since the late 1970’s, which seeks refinement of the celebrated Normal Subgroup Theorem of Margulis (hereafter abbreviated NST). Although Margulis’ NST is stated and proved in the context of (higher rank) irreducible lattices in products of simple algebraic groups over local fields, by Margulis’ arithmeticity theorem we may and shall work solely in the framework of (S-)arithmetic groups. One departure point for the Margulis-Zimmer conjecture is the phenomenon that while all higher rank S-arithmetic groups are uniformly treated by the NST, there is a notable difference in the structure of subgroups which are commensurated, rather than normalized, by the ambient arithmetic group. For example, the group SLn(Z[ p ]) commensurates its subgroup SLn(Z), while the latter commensurates no apparent infinite, infinite index subgroup of its own. The obvious generalization of this example, which by Margulis arithmeticity theorem and with the aid of the restriction of scalars functor is the most general one, goes as follows: