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Norm rigidity for arithmetic and profinite groups

Leonid Polterovich, Yehuda Shalom, Zvi Shem-Tov

TL;DR

This work establishes a non‑abelian rigidity phenomenon for conjugation‑invariant norms on arithmetic and profinite groups, introducing the dichotomy property (every non‑trivial norm is either discrete or precompact) and proving it holds for ${\rm E}_n(A)$ when bounded elementary generation is present and every nonzero ideal of $A$ has finite index. It develops a dual compact setting via norm completeness, showing norm‑complete compact groups are either profinite or (p‑adic/real) semisimple Lie groups, and proves that ${\rm SL}_n(A)$ over compact rings with the same ideal condition are norm complete. The paper also provides concrete applications, including a finitely presented meager group from a non‑split central extension and automatic continuity results for homomorphisms, and it outlines connections to Margulis’ normal subgroup theorem and Nikolov–Segal theory, with several open problems guiding future work on higher rank and non‑residually finite contexts. Overall, it links norm rigidity to deep rigidity phenomena in arithmetic and profinite groups, with implications for sofic approximations and central extensions. The results unify discrete and compact perspectives through bounded generation and finite‑index phenomena, offering a robust framework for understanding how metric completeness constraints shape algebraic structure.

Abstract

Let $A$ be a commutative ring, and assume every non-trivial ideal of $A$ has finite-index. We show that if ${\rm{SL}}_n(A)$ has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If $G$ is any group satisfying this dichotomy we say that $G$ has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.

Norm rigidity for arithmetic and profinite groups

TL;DR

This work establishes a non‑abelian rigidity phenomenon for conjugation‑invariant norms on arithmetic and profinite groups, introducing the dichotomy property (every non‑trivial norm is either discrete or precompact) and proving it holds for when bounded elementary generation is present and every nonzero ideal of has finite index. It develops a dual compact setting via norm completeness, showing norm‑complete compact groups are either profinite or (p‑adic/real) semisimple Lie groups, and proves that over compact rings with the same ideal condition are norm complete. The paper also provides concrete applications, including a finitely presented meager group from a non‑split central extension and automatic continuity results for homomorphisms, and it outlines connections to Margulis’ normal subgroup theorem and Nikolov–Segal theory, with several open problems guiding future work on higher rank and non‑residually finite contexts. Overall, it links norm rigidity to deep rigidity phenomena in arithmetic and profinite groups, with implications for sofic approximations and central extensions. The results unify discrete and compact perspectives through bounded generation and finite‑index phenomena, offering a robust framework for understanding how metric completeness constraints shape algebraic structure.

Abstract

Let be a commutative ring, and assume every non-trivial ideal of has finite-index. We show that if has bounded elementary generation then every conjugation-invariant norm on it is either discrete or precompact. If is any group satisfying this dichotomy we say that has the \emph{dichotomy property}. We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research.

Paper Structure

This paper contains 23 sections, 35 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Norms on groups and some basic constructions.
  3. The main results -- discrete setting.
  4. The main results -- dual compact setting.
  5. An example of a finitely-presented meager group with possible application to the theory of approximable groups.
  6. Plan of the paper.
  7. Acknowledgments.
  8. Norm rigidity for abstract groups
  9. Proof of Theorem \ref{['mainthm']} \ref{['main11']}
  10. Preparations for the proof of Theorem \ref{['mainthm']} \ref{['main21']} -- Behavior under passage to finite-index subgroups, and the weak dichotomy property
  11. Proof of the second part of Theorem \ref{['mainthm']}
  12. Proof of Theorem \ref{['y51']}
  13. Norm rigidity for compact groups
  14. Generalities
  15. Proof of Proposition \ref{['auto']}
  16. ...and 8 more sections

Key Result

Proposition 1.5

Let $G$ be a residually finite group. Assume that $G$ has the dichotomy property. Let $N$ be a normal subgroup of $G$. Then either $N$ has finite index in $G$ or $N$ is finite.

Theorems & Definitions (73)

  • Definition 1.3
  • Remark 1.4
  • Proposition 1.5
  • proof
  • Example 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 1.10
  • Proposition 1.11
  • ...and 63 more