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

By Yehuda Shalom, George A. Willis. 2009

81 citations recorded by the source corpus.

The paper establishes a substantial number of cases of a conjecture regarding commensurated subgroups of S-arithmetic groups made by Margulis and Zimmer in the late 1970s. New results in the structure theory of totally disconnected groups are established along the way and are of independent interest. Other ideas in the argument motivate a sweeping conjecture, presented in the last section of the paper, which naturally unifies in an adelic setting deep results and fundamental conjectures in the rigidity theory of arithmetic groups.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1e605b0d9672b894fa8864a6049ea934a04b520c

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Citations in the corpus

Stabilizers of Ergodic Actions of Lattices and Commensurators (2013, unknown): “out the relative proﬁnite completion of commensurated subgroups. This construction has appeared in the study of commensurated subgroups [Sch80], [Tza00], [Tza03], [CM09] and the reader is referred to [SW13] and [Cre11] for more information and proofs of the following basic facts. Deﬁnition 6.2. Let ”
LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS (2014, background): “the Margulis–Zimmer conjecture according to which a lattice in a connected centreless simple Lie group of rank ≥ 2 has no commensurated subgroup other than the ﬁnite or the ﬁnite index subgroups (see [57] for a thorough discussion of that conjecture as well as partial results on non-uniform lattices”