15 August 2023
Wouter van Limbeek
Greenberg-Shalom conjecture predicts discrete subgroups with dense commensurators are lattices
Would imply nonexistence of irreducible surface groups in products of Lie groups
Also would imply coherence of many arithmetic groups like SL(2,Z[1/p])
And bounded generation of irreducible lattices by few elements
Proves a case of the Margulis-Zimmer conjecture on commensurated subgroups
Commensurators and lattices in semisimple Lie groups
This paper discusses implications of a conjecture of Greenberg and Shalom that discrete subgroups of semisimple Lie groups with dense commensurators are lattices. The conjecture is shown to have surprising consequences for problems on existence of irreducible discrete subgroups, coherence of arithmetic groups, ranks of lattices, and the Margulis-Zimmer conjecture.
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