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Metric ultraproducts of groups -- simplicity, perfectness and torsion

Jakub Gismatullin, Krzysztof Majcher, Martin Ziegler

Abstract

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

Metric ultraproducts of groups -- simplicity, perfectness and torsion

Abstract

We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.

Paper Structure

This paper contains 23 sections, 36 theorems, 95 equations.

Table of Contents

  1. Metric ultraproducts
  2. Notation
  3. New compactness theorem for metric ultraproducts of groups
  4. Metrically internal sets
  5. New compactness theorem
  6. Corollaries of compactness theorem
  7. Square of finite subcover
  8. Bounded generation, torsion, perfectness and simplicity
  9. Bounded generation
  10. Uniform perfectness
  11. Uniform torsion
  12. Uniform and bounded simplicity
  13. Simplicity
  14. Characterization of simplicity
  15. A construction based on a single group
  16. ...and 8 more sections

Key Result

Theorem 1

thm:sim_b (bounded case) The following conditions are equivalent.

Theorems & Definitions (103)

  • Conjecture
  • Theorem
  • Theorem
  • Corollary
  • Corollary
  • Corollary
  • Conjecture
  • Example
  • Theorem
  • Theorem
  • ...and 93 more