[Paper Review] Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity

Yehuda Shalom, George A. Willis|ArXiv.org|Nov 10, 2009

Advanced Algebra and Geometry37 references4 citations

TL;DR

This paper establishes adelic rigidity for arithmetic groups by introducing the outer commensurator-normalizer property, proving that commensurated subgroups in higher-rank $S$-arithmetic groups are almost normal. It resolves the Margulis-Zimmer commensurated subgroup problem affirmatively in characteristic zero, showing that all such subgroups are standard or finite, and links this to superrigidity and Deligne's central extensions via a local-global principle.

ABSTRACT

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.

Motivation & Objective

• To resolve the Margulis-Zimmer commensurated subgroup problem (CmSP) for higher-rank $S$-arithmetic groups in characteristic zero.
• To establish a strong fixed-point property—outer commensurator-normalizer—enabling reduction of the CmSP to Margulis' Normal Subgroup Theorem as a black box.
• To unify the CmSP with the Congruence Subgroup Problem (CSP) and superrigidity via a local-global principle for homomorphisms into central extensions of algebraic groups.
• To clarify the role of Deligne’s central extension in determining when $S$-arithmetic groups lift to perfect central extensions, especially over finite index subgroups.
• To provide a criterion for the existence of non-trivial homomorphisms from $S$-arithmetic groups into connected, locally isomorphic groups via the structure of $ ilde{G}^{(nc)}_ ho$.

Proposed method

• Introduce the outer commensurator-normalizer property as a commensurability-stable strengthening of the inner property, ensuring that commensurated subgroups are almost normalized under any homomorphism.
• Use Margulis’ Normal Subgroup Theorem as a black box, reducing the CmSP to proving that the outer property holds for $S$-arithmetic groups.
• Apply superrigidity and projectivization techniques to analyze homomorphisms from $S$-arithmetic groups into central extensions of products of local groups $G_v = f G(K_v)$.
• Leverage Deligne’s construction of central extensions of algebraic groups over $K$-rational points to analyze lifting obstructions and splitting conditions.
• Use the structure of the group $ ilde{G}^{(nc)}_ ho$ from Proposition 7.8 to determine whether a given central extension admits a non-trivial lift of an $S$-arithmetic group.
• Establish a duality: a non-trivial homomorphism exists into $H$ if and only if $H$ is a quotient of $ ilde{G}^{(nc)}_ ho$, unifying the CmSP and CSP through a local-global principle.

Experimental results

Research questions

• RQ1 Does every commensurated subgroup of a higher-rank $S$-arithmetic group $Γ$ arise as a standard $S'$-arithmetic subgroup or be finite?
• RQ2 Can the outer commensurator-normalizer property be established for $S$-arithmetic groups, ensuring stability under finite index subgroups?
• RQ3 Under what conditions does a central extension of a product of local groups $G_v$ admit a non-trivial lift of an $S$-arithmetic group?
• RQ4 How do Deligne’s central extensions interact with the structure of $S$-arithmetic groups, particularly over finite index subgroups?
• RQ5 Is there a precise local-global correspondence determining whether a homomorphism from an $S$-arithmetic group to a connected, locally isomorphic group exists?

Key findings

• The Margulis-Zimmer commensurated subgroup problem has a positive solution in characteristic zero: every commensurated subgroup of a higher-rank $S$-arithmetic group is either finite or standard.
• The outer commensurator-normalizer property holds for $S$-arithmetic groups, ensuring that commensurated subgroups are almost normalized under any homomorphism, a property stable under finite index subgroups.
• For $\Gamma = SL_n(\mathbb{Z}[1/p])$, no finite index subgroup of the preimage $\tilde{\Gamma} < \widetilde{SL_n(\mathbb{R})}$ is center-free or residually finite, but over finite index subgroups of $SL_n(\mathbb{Z})$, Deligne’s extension splits.
• Over a finite index subgroup $\Gamma'$ of $SL_n(\mathbb{Z})$, the central extension $\pi: \widetilde{SL_n(\mathbb{R})} \to SL_n(\mathbb{R})$ admits a splitting, implying $\tilde{\Gamma}'$ is virtually center-free and residually finite.
• A non-trivial homomorphism from an $S$-arithmetic group $\Gamma$ to a connected, locally isomorphic group $H$ exists if and only if $H$ is a quotient of $\tilde{G}^{(nc)}_\Gamma$, establishing a precise duality between lifting obstructions and existence.