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Conjugacy width in uniform higher rank arithmetic groups of orthogonal type

Nir Avni, Chen Meiri

TL;DR

This paper proves that, under GRH, many uniform higher-rank $S$-arithmetic groups of orthogonal type (anisotropic Spin groups with $n\ge17$) exhibit finite conjugacy width, linking width to local conjugacy widths and CSP data. The authors develop a non-standard (ultrapower) framework, define congruence topologies on ultraproducts, and analyze discrete metaplectic extensions via Steinberg symbols to control commutator/width behavior. A key strategy is translating width questions into CSP-type problems in non-standard models, using approximation in tori and a thorough analysis of split spin groups to achieve finite kernels in metaplectic extensions. The work further connects FCW to CSP, yields bounded generation-like consequences in some cases, and has model-theoretic and stability implications, with unconditional BG results anticipated in related work AM25. Overall, the paper advances a non-classical route to understand width phenomena in high-rank anisotropic arithmetic groups, with wide-ranging consequences for verbal width, model theory, and stability.

Abstract

We study widths of conjugacy classes in anisotropic higher rank $S$-arithmetic groups of orthogonal type. Assuming the GRH, we prove that many such groups have bounded conjugacy width. For example, this holds if the degree is greater or equal to 17 and $S$ contains a non-archimedean place. To the best of our knowledge, this is the first boundedness result proved for anisotropic groups. The proof uses ideas from the Congruence Subgroup Problem. In particular, we define and compute a non standard version of the metaplectic kernel. Conversely, we prove that a quantitative bound on the width of conjugacy classes implies the CSP. The machinery we develop can also be used for other width questions. For example, in \cite{AM25} we prove, unconditional on GRH, new cases of bounded generation of arithmetic groups.

Conjugacy width in uniform higher rank arithmetic groups of orthogonal type

TL;DR

This paper proves that, under GRH, many uniform higher-rank -arithmetic groups of orthogonal type (anisotropic Spin groups with ) exhibit finite conjugacy width, linking width to local conjugacy widths and CSP data. The authors develop a non-standard (ultrapower) framework, define congruence topologies on ultraproducts, and analyze discrete metaplectic extensions via Steinberg symbols to control commutator/width behavior. A key strategy is translating width questions into CSP-type problems in non-standard models, using approximation in tori and a thorough analysis of split spin groups to achieve finite kernels in metaplectic extensions. The work further connects FCW to CSP, yields bounded generation-like consequences in some cases, and has model-theoretic and stability implications, with unconditional BG results anticipated in related work AM25. Overall, the paper advances a non-classical route to understand width phenomena in high-rank anisotropic arithmetic groups, with wide-ranging consequences for verbal width, model theory, and stability.

Abstract

We study widths of conjugacy classes in anisotropic higher rank -arithmetic groups of orthogonal type. Assuming the GRH, we prove that many such groups have bounded conjugacy width. For example, this holds if the degree is greater or equal to 17 and contains a non-archimedean place. To the best of our knowledge, this is the first boundedness result proved for anisotropic groups. The proof uses ideas from the Congruence Subgroup Problem. In particular, we define and compute a non standard version of the metaplectic kernel. Conversely, we prove that a quantitative bound on the width of conjugacy classes implies the CSP. The machinery we develop can also be used for other width questions. For example, in \cite{AM25} we prove, unconditional on GRH, new cases of bounded generation of arithmetic groups.
Paper Structure (51 sections, 93 theorems, 134 equations)

This paper contains 51 sections, 93 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Applications
  4. Bounded Generation
  5. Verbal width
  6. Model Theory of arithmetic groups
  7. Stability
  8. Sketch of the proof: Verbal width and CSP in non-standard models
  9. Acknowledgements
  10. Notations
  11. Congruence topologies on ultraproducts
  12. Internal sets and ultraproduct topologies
  13. Topological groups---general properties
  14. Internal ideals and sets of ideals
  15. Valuations
  16. ...and 36 more sections

Key Result

Theorem 1.1.1

Assume the Generalized Riemann Hypothesis holds. Let $K$ be a number field, let $S$ be a finite set of places of $K$ containing all archimedean places, let $n \geq 17$, let $f_a$ be an anisotropic quadratic form on $K^n$, and let $\Gamma \subseteq \mathop{\mathrm{Spin}}\nolimits_{f_a}(K)$ be an $S$- Then $\Gamma$ has finite conjugacy width. Moreover, there is a constant $C$ such that, for every $\

Theorems & Definitions (228)

  • Theorem 1.1.1
  • Conjecture 1.1.2
  • Theorem 1.1.3
  • Remark 1.1.4
  • Remark 1.1.5
  • Definition 1.2.1
  • Corollary 1.2.2
  • Conjecture 1.2.3
  • Remark 1.2.4
  • Remark 1.2.5
  • ...and 218 more