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Commensurators and lattices in semisimple Lie groups

Published on:

15 August 2023

Primary Category:

Group Theory

Paper Authors:

Nic Brody,

David Fisher,

Mahan Mj,

Wouter van Limbeek


Key Details

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

AI generated summary

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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