[Le sous-groupe 3-colorable et les sous-groupes maximaux du groupe de Thompson ]
Vaughan Jones a introduit et étudié un sous-groupe du groupe de Thompson dit le sous-groupe 3-colorable, apparu naturellement dans son travail sur les représentations unitaires de . Ren a montré que ce sous-groupe est isomorphe au groupe de Brown–Thompson. Ici, nous poursuivons l’étude du sous-groupe 3-colorable et démontrons que la représentation quasi-régulière de qui lui est associée est irréductible. Nous démontrons de plus que la préimage de par un certain endomorphisme injectif de est contenue dans trois sous-groupes maximaux de que nous construisons explicitement. Ces sous-groupes maximaux sont d’indice infini et sont des nouveaux exemples dans la liste des sous-groupes maximaux d’indice infini connus dans , tels les sous-groupes paraboliques fixant un point de l’intervalle , le sous-groupe orienté introduit par Jones, et les exemples construits par Golan.
In his work on representations of Thompson’s group , Vaughan Jones defined and studied the -colorable subgroup of . Later, Ren showed that it is isomorphic to the Brown–Thompson group . In this paper we continue with the study of the -colorable subgroup and prove that the quasi-regular representation of associated with the -colorable subgroup is irreducible. We show in addition that the preimage of under a certain injective endomorphism of is contained in three (explicit) maximal subgroups of of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of , namely the parabolic subgroups that fix a point in , (up to isomorphism) the Jones’ oriented subgroup , and the explicit examples found by Golan.
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Keywords: Thompson groups, Brown–Thompson groups, irreducible unitary representations, Jones representation, infinite index maximal subgroups, stabilizer subgroups, chromatic polynomial, closed subgroups.
Mot clés : Groupes de Thompson, groupes de Brown–Thompson, représentation unitaire irréductible de groupe, représentations de Jones, sous-groupe maximal d’indice infini, sous-groupe stabilisateur, polynôme chromatique, sous-groupe fermé.
Aiello, Valeriano 1 ; Nagnibeda, Tatiana 2
CC-BY-ND 4.0
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author = {Aiello, Valeriano and Nagnibeda, Tatiana},
title = {On the $3$-colorable subgroup $\protect \mathcal{F}$ and maximal subgroups of {Thompson{\textquoteright}s} group $F$},
journal = {Annales de l'Institut Fourier},
pages = {783--828},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {73},
number = {2},
year = {2023},
doi = {10.5802/aif.3555},
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url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3555/}
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Aiello, Valeriano; Nagnibeda, Tatiana. On the $3$-colorable subgroup $\protect \mathcal{F}$ and maximal subgroups of Thompson’s group $F$. Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 783-828. doi : 10.5802/aif.3555. https://aif.centre-mersenne.org/articles/10.5802/aif.3555/
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