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Rigidity and automorphisms of groups constructed using Jones' technology

Christian De Nicola Larsen

Abstract

Jones' technology, developed by Vaughan Jones during his exploration of the connections between conformal field theory and subfactors, is a powerful mechanism for generating actions of groups coming from categories, notably Richard Thompson's groups $F \subseteq T \subseteq V$. We give a structure theorem for the isomorphisms between split extensions of Thompson's group $V$ arising from Jones' technology, generalising results of Brothier. Using this structure theorem, we classify a family of unrestricted, twisted permutational wreath products up to isomorphism, and decompose their automorphism groups in the untwisted case. These unrestricted wreath products arise from applying Jones' technology to contravariant monoidal functors. In contrast, using covariant functors, Brothier constructed a large class of restricted wreath products, classified them up to isomorphism, and completely described their automorphism groups. Our work broadens Brothier's findings and highlights the duality between groups constructed using covariant and contravariant functors.

Rigidity and automorphisms of groups constructed using Jones' technology

Abstract

Jones' technology, developed by Vaughan Jones during his exploration of the connections between conformal field theory and subfactors, is a powerful mechanism for generating actions of groups coming from categories, notably Richard Thompson's groups . We give a structure theorem for the isomorphisms between split extensions of Thompson's group arising from Jones' technology, generalising results of Brothier. Using this structure theorem, we classify a family of unrestricted, twisted permutational wreath products up to isomorphism, and decompose their automorphism groups in the untwisted case. These unrestricted wreath products arise from applying Jones' technology to contravariant monoidal functors. In contrast, using covariant functors, Brothier constructed a large class of restricted wreath products, classified them up to isomorphism, and completely described their automorphism groups. Our work broadens Brothier's findings and highlights the duality between groups constructed using covariant and contravariant functors.
Paper Structure (20 sections, 61 theorems, 264 equations)

This paper contains 20 sections, 61 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Other results on wreath products
  4. Covariant vs. contravariant functors
  5. Plan of the article
  6. Preliminaries
  7. Localisation
  8. Calculus of fractions
  9. Thompson groupoids
  10. Jones' technology
  11. Thompson's groups
  12. Groups constructed from a Jones action
  13. Isomorphism structure
  14. Obvious isomorphisms
  15. Contravariant Jones' technology
  16. ...and 5 more sections

Key Result

Theorem A1

Suppose that $\alpha: \Gamma \to \Gamma^2$ and $\widetilde{\alpha}: \widetilde{\Gamma} \to \widetilde{\Gamma}^2$ are group morphisms, and that $\theta: G(\alpha) \to G(\widetilde{\alpha})$ is an isomorphism of groups. Then there exists an isomorphism $\kappa^0: K(\alpha) \to K(\widetilde{\alpha})$, and Moreover, the map is an isomorphism of groups.

Theorems & Definitions (144)

  • Theorem A1
  • Theorem A2
  • Theorem B1
  • Theorem B2
  • Theorem B3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 134 more