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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).
Koberda, Thomas 1 ; Mj, Mahan 2
@article{10_2140_agt_2024_24_2149,
author = {Koberda, Thomas and Mj, Mahan},
title = {Commensurators of thin normal subgroups and abelian quotients},
journal = {Algebraic and Geometric Topology},
pages = {2149--2170},
publisher = {mathdoc},
volume = {24},
number = {4},
year = {2024},
doi = {10.2140/agt.2024.24.2149},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2149/}
}
TY - JOUR AU - Koberda, Thomas AU - Mj, Mahan TI - Commensurators of thin normal subgroups and abelian quotients JO - Algebraic and Geometric Topology PY - 2024 SP - 2149 EP - 2170 VL - 24 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2149/ DO - 10.2140/agt.2024.24.2149 ID - 10_2140_agt_2024_24_2149 ER -
%0 Journal Article %A Koberda, Thomas %A Mj, Mahan %T Commensurators of thin normal subgroups and abelian quotients %J Algebraic and Geometric Topology %D 2024 %P 2149-2170 %V 24 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2024.24.2149/ %R 10.2140/agt.2024.24.2149 %F 10_2140_agt_2024_24_2149
Koberda, Thomas; Mj, Mahan. Commensurators of thin normal subgroups and abelian quotients. Algebraic and Geometric Topology, Tome 24 (2024) no. 4, pp. 2149-2170. doi: 10.2140/agt.2024.24.2149
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