On the 3-colorable subgroup and maximal subgroups of Thompson’s group F
Annales de l'Institut Fourier, Volume 73 (2023) no. 2, pp. 783-828.

In his work on representations of Thompson’s group F, Vaughan Jones defined and studied the 3-colorable subgroup of F. Later, Ren showed that it is isomorphic to the Brown–Thompson group F 4 . In this paper we continue with the study of the 3-colorable subgroup and prove that the quasi-regular representation of F associated with the 3-colorable subgroup is irreducible. We show in addition that the preimage of under a certain injective endomorphism of F is contained in three (explicit) maximal subgroups of F of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of F, namely the parabolic subgroups that fix a point in (0,1), (up to isomorphism) the Jones’ oriented subgroup F , and the explicit examples found by Golan.

Vaughan Jones a introduit et étudié un sous-groupe du groupe de Thompson F dit le sous-groupe 3-colorable, apparu naturellement dans son travail sur les représentations unitaires de F. Ren a montré que ce sous-groupe est isomorphe au groupe F 4 de Brown–Thompson. Ici, nous poursuivons l’étude du sous-groupe 3-colorable et démontrons que la représentation quasi-régulière de F qui lui est associée est irréductible. Nous démontrons de plus que la préimage de par un certain endomorphisme injectif de F est contenue dans trois sous-groupes maximaux de F 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 F, tels les sous-groupes paraboliques fixant un point de l’intervalle (0,1), le sous-groupe orienté F introduit par Jones, et les exemples construits par Golan.

Published online:
DOI: 10.5802/aif.3555
Classification: 20F65, 20E28, 22D10
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

1 Mathematisches Institut Universität Bern Alpeneggstrasse 22 3012 Bern (Switzerland)
2 Section de mathématiques, Université de Genève Rue du Conseil-Général, 7-9 2005 Genève (Swtizerland)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Volume 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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