Commensurators of thin normal subgroups and abelian quotients

Koberda, Thomas; Mj, Mahan

doi:10.2140/agt.2024.24.2149

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Commensurators of thin normal subgroups and abelian quotients

Thomas Koberda and Mahan Mj

Algebraic & Geometric Topology 24 (2024) 2149–2170

DOI: 10.2140/agt.2024.24.2149

Abstract

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. Let $K < Γ < G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank-one simple Lie group $G$ , such that the quotient $Q = Γ ∕ K$ is infinite. We show that the commensurator of $K$ in $G$ is discrete, provided that $Q$ admits a surjective homomorphism to $ℤ$ . In this case, we also show that the commensurator of $K$ contains the normalizer of $K$ with finite index. We thus vastly generalize a 2021 result of the authors, which showed that many natural normal subgroups of ${PSL}_{2} (ℤ)$ have discrete commensurator in ${PSL}_{2} (ℝ)$ .

Keywords

commensurator, Hodge theory, coarse preservation of lines, arithmetic lattice, thin subgroup

Mathematical Subject Classification

Primary: 22E40

Secondary: 20F65, 20F67, 57M50

References

Publication

Received: 9 May 2022

Revised: 16 April 2023

Accepted: 12 May 2023

Published: 16 July 2024

Authors

Thomas Koberda
Department of Mathematics University of Virginia Charlottesville, VA United States
https://sites.google.com/view/koberdat
Mahan Mj
School of Mathematics Tata Institute of Fundamental Research Mumbai India
http://www.math.tifr.res.in/~mahan

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Volume 24, issue 4 (2024)