Greenberg-Shalom's Commensurator Hypothesis and Applications

2023/08/15 by Brody, Nic, Fisher, David, Mj, Mahan +1 · 2 citations
#22E40 #53C35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2308.07785

Abstract

We discuss many surprising implications of a positive answer to a question raised in some cases by Greenberg in the `70s and more generally by Shalom in the early 2000s. We refer to this positive answer as the Greenberg-Shalom hypothesis. This hypothesis then says that any infinite discrete subgroup of a semisimple Lie group with dense commensurator is a lattice in a product of some factors. For some applications it is natural to extend the hypothesis to cover semisimple algebraic groups over other fields as well.

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• Commensurators of normal subgroups of lattices
• Indiscrete Common Commensurators
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• Abstract commensurators of lattices in Lie groups
• Weakly commensurable groups, with applications to differential geometry