Haagerup property for wreath products constructed with Thompson's groups

Brothier, Arnaud

doi:10.48550/arxiv.1906.03789

Cited by 5 publications

(12 citation statements)

References 22 publications

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“…With this method the author proved that a large class of semidirect products had the Haagerup property. In particular, all wreath products ⊕ Q 2 Γ V , with Γ any group having the Haagerup property and V Q 2 the classical action of Thompson group V on the dyadic rational, have the Haagerup property [Bro19a]. This provided, using a result of Cornulier, the first examples of finitely presented wreath products having the Haagerup property for a nontrivial reason (i.e.…”

Section: Introductionmentioning

confidence: 93%

“…If Φ is a covariant functor and D is the category of Hilbert spaces with isometries for morphisms, then X Φ is a preHilbert space and the Jones action π Φ can be extended into a unitary representation of G C on the completion of X Φ . This provides a wonderful machine for constructing unitary representations and matrix coefficients, see [Jon19a,BJ19b,BJ19a,Bro19a,ABC19]. If Φ : C → Gr is a covariant functor where Gr is the category of groups, then X Φ is a group and the Jones action π Φ : G C X Φ is an action by automorphisms on this group.…”

Section: Preliminariesmentioning

confidence: 99%

“…In particular, the fraction group of the category of binary forests (at the object 1) is Thompson group F : elements of F are described by (an equivalence class of) a pair of trees having the same number of leaves. The larger groups T and V have similar descriptions but with trees having their leaves decorated by natural numbers corresponding to permutations, see [Bel04,Bro19a]. Jones made the crucial observation that, if we consider any functor starting from such a category (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…• A functor Φ : C → Gr ending in the category of groups gives an action G C K of the fraction group G C associated to the source category C on a limit group K. The author made the key observation that the semidirect product K G C is again a fraction group and provided an explicit description of the category inducing this group [Bro19a]. Jones framework can be then reapplied to this semidirect product K G C for constructing unitary representations and computing matrix coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Classification of Thompson related groups arising from Jones technology I

Brothier¹

2020

Preprint

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In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful framework for constructing actions of particular groups arising from categories such as Richard Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product. Those semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of groups having the Haagerup property. Moreover, they naturally appear in certain field theories as being generated by local and global symmetries.We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a second article. The groups obtained can be described in terms of permutational restricted twisted wreath products containing the larger Thompson group. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.À la mémoire de Vaughan Jones

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Section: Introductionmentioning

confidence: 93%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of Thompson related groups arising from Jones technology I

Brothier¹

2020

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“…[OS19,KK18]. The actions in the context of groups have just been used by one of the authors in a very different context [Bro19]. The article is structured as follows:…”

Section: Introductionmentioning

confidence: 99%

Canonical quantization of 1+1-dimensional Yang-Mills theory: An operator-algebraic approach

Brothier¹,

Stottmeister²

2019

Preprint

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We present a mathematically rigorous canonical quantization of Yang-Mills theory in 1+1 dimensions (YM1`1) by operator-algebraic methods. The latter are based on Hamiltonian lattice gauge theory and multi-scale analysis via inductive limits of C ˚-algebras which are applicable in arbitrary dimensions. The major step, restricted to one spatial dimension, is the explicitly construction of the spatially-localized von Neumann algebras of time-zero fields in the time gauge in representations associated with scaling limits of Gibbs states of the Kogut-Susskind Hamiltonian. We relate our work to existing results about YM1`1 and its counterpart in Euclidean quantum field theory (YM2). In particular, we show that the operator-algebraic approach offers a unifying perspective on results about YM1`1 obtained by Dimock as well as Driver and Hall, especially regarding the existence of dynamics. Although our constructions work for non-abelian gauge theory, we obtain the most explicit results in the abelian case by applying the results of our recent companion article. In view of the latter, we also discuss relations with the construction of unitary representations of Thompson's groups by Jones. To understand the scaling limits arising from our construction, we explain our findings via a rigorous adaptation of the Wilson-Kadanoff renormalization group, which connects our construction with the multi-scale entanglement renormalization ansatz (MERA). Finally, we discuss potential generalizations and extensions to higher dimensions (d `1 ě 3).

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Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

Brothier

Stottmeister

2019

Commun. Math. Phys.

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Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group G, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of G over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group T as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of G we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian G, we provide a very explicit description of our algebras. For a single measure on the dual of G, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For G " S the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group T , for geometrically motivated choices of families of heat-kernel states.

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Haagerup property for wreath products constructed with Thompson's groups

Cited by 5 publications

References 22 publications

Classification of Thompson related groups arising from Jones technology I

Classification of Thompson related groups arising from Jones technology I

Canonical quantization of 1+1-dimensional Yang-Mills theory: An operator-algebraic approach

Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

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