On the 3-colorable subgroup and maximal subgroups of Thompson’s group F
Annales de l'Institut Fourier, Online first, 46 p.

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.

Online First:
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.
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)
     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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3555},
     language = {en},
     note = {Online first},
AU  - Aiello, Valeriano
AU  - Nagnibeda, Tatiana
TI  - On the $3$-colorable subgroup $\protect \mathcal{F}$ and maximal subgroups of Thompson’s group $F$
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3555
DO  - 10.5802/aif.3555
LA  - en
ID  - AIF_0__0_0_A113_0
ER  - 
%0 Unpublished Work
%A Aiello, Valeriano
%A Nagnibeda, Tatiana
%T On the $3$-colorable subgroup $\protect \mathcal{F}$ and maximal subgroups of Thompson’s group $F$
%J Annales de l'Institut Fourier
%D 2022
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3555
%R 10.5802/aif.3555
%G en
%F AIF_0__0_0_A113_0
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, Online first, 46 p.

[1] Aiello, Valeriano On the Alexander theorem for the oriented Thompson group F , Algebr. Geom. Topol., Volume 20 (2020) no. 1, pp. 429-438 | DOI | MR | Zbl

[2] Aiello, Valeriano; Baader, Sebastian Arborescence of positive Thompson links, Pacific J. Math., Volume 316 (2022) no. 2, pp. 237-248 | DOI | MR | Zbl

[3] Aiello, Valeriano; Brothier, Arnaud; Conti, Roberto Jones representations of Thompson’s group F arising from Temperley-Lieb-Jones algebras, Int. Math. Res. Not. IMRN (2021) no. 15, pp. 11209-11245 | DOI | MR | Zbl

[4] Aiello, Valeriano; Conti, Roberto Graph polynomials and link invariants as positive type functions on Thompson’s group F, J. Knot Theory Ramifications, Volume 28 (2019) no. 2, 1950006, 17 pages | DOI | MR | Zbl

[5] Aiello, Valeriano; Conti, Roberto The Jones polynomial and functions of positive type on the oriented Jones–Thompson groups F and T , Complex Anal. Oper. Theory, Volume 13 (2019) no. 7, pp. 3127-3149 https://doi-org.proxy.library.vanderbilt.edu/10.1007/s11785-018-0866-6 | DOI | MR | Zbl

[6] Aiello, Valeriano; Conti, Roberto; Jones, Vaughan F. R. The Homflypt polynomial and the oriented Thompson group, Quantum Topol., Volume 9 (2018) no. 3, pp. 461-472 https://doi-org.proxy.library.vanderbilt.edu/10.4171/QT/112 | DOI | MR | Zbl

[7] Aiello, Valeriano; Jones, Vaughan F. R. On spectral measures for certain unitary representations of R. Thompson’s group F, J. Funct. Anal., Volume 280 (2021) no. 1, 108777, 27 pages https://doi-org.proxy.library.vanderbilt.edu/10.1016/j.jfa.2020.108777 | DOI | MR | Zbl

[8] Aiello, Valeriano; Nagnibeda, Tatiana On the oriented Thompson subgroup F 3 and its relatives in higher Brown–Thompson groups, J. Algebra Appl., Volume 21 (2022) no. 7, 2250139, 21 pages | DOI | MR | Zbl

[9] Belk, James Thompson’s group F (2007) (https://arxiv.org/abs/0708.3609)

[10] Belk, James; Matucci, Francesco Conjugacy and dynamics in Thompson’s groups, Geom. Dedicata, Volume 169 (2014), pp. 239-261 https://doi-org.proxy.library.vanderbilt.edu/10.1007/s10711-013-9853-2 | DOI | MR | Zbl

[11] Bleak, Collin; Wassink, Bronlyn Finite index subgroups of R. Thompson’s group F (2007) (http://arxiv.org/abs/0711.1014)

[12] Brothier, Arnaud; Jones, Vaughan F. R. Pythagorean representations of Thompson’s groups, J. Funct. Anal., Volume 277 (2019) no. 7, pp. 2442-2469 https://doi-org.proxy.library.vanderbilt.edu/10.1016/j.jfa.2019.02.009 | DOI | MR

[13] Brown, Kenneth S. Finiteness properties of groups, J. Pure Appl. Algebra, Volume 44 (1987) no. 1-3, pp. 45-75 https://doi-org.proxy.library.vanderbilt.edu/10.1016/0022-4049(87)90015-6 | DOI | MR | Zbl

[14] Cannon, James W.; Floyd, William J.; Parry, Walter R. Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), Volume 42 (1996) no. 3-4, pp. 215-256 | MR | Zbl

[15] Dudko, Artem; Medynets, Konstantin Finite factor representations of Higman–Thompson groups, Groups Geom. Dyn., Volume 8 (2014) no. 2, pp. 375-389 https://doi-org.proxy.library.vanderbilt.edu/10.4171/GGD/230 | DOI | MR | Zbl

[16] Francoeur, Dominik On the stabilisers of points in groups with micro-supported actions (2020) (to appear in J. Group Theory)

[17] Golan, Gili The generation problem in Thompson group F (2016) (https://arxiv.org/abs/1608.02572)

[18] Golan, Gili; Sapir, Mark On Jones’ subgroup of R. Thompson group F, J. Algebra, Volume 470 (2017), pp. 122-159 https://doi-org.proxy.library.vanderbilt.edu/10.1016/j.jalgebra.2016.09.001 | DOI | MR | Zbl

[19] Golan, Gili; Sapir, Mark On subgroups of R. Thompson’s group F, Trans. Amer. Math. Soc., Volume 369 (2017) no. 12, pp. 8857-8878 https://doi-org.proxy.library.vanderbilt.edu/10.1090/tran/6982 | DOI | MR | Zbl

[20] Golan, Gili; Sapir, Mark On the stabilizers of finite sets of numbers in the R. Thompson group F, Algebra i Analiz, Volume 29 (2017) no. 1, pp. 70-110 https://doi-org.proxy.library.vanderbilt.edu/10.1090/spmj/1482 | DOI | MR | Zbl

[21] Higman, Graham Finitely presented infinite simple groups, Notes on Pure Mathematics, Australian National University, Department of Mathematics, I.A.S., Canberra, 1974 no. 8, vii+82 pages | MR

[22] Jones, V. F. R. Planar algebras, I, New Zealand J. Math., Volume 52 (2021 [2021–2022]), pp. 1-107 | DOI | MR | Zbl

[23] Jones, Vaughan F. R. Some unitary representations of Thompson’s groups F and T, J. Comb. Algebra, Volume 1 (2017) no. 1, pp. 1-44 https://doi-org.proxy.library.vanderbilt.edu/10.4171/JCA/1-1-1 | DOI | MR | Zbl

[24] Jones, Vaughan F. R. A no-go theorem for the continuum limit of a periodic quantum spin chain, Comm. Math. Phys., Volume 357 (2018) no. 1, pp. 295-317 https://doi-org.proxy.library.vanderbilt.edu/10.1007/s00220-017-2945-3 | DOI | MR | Zbl

[25] Jones, Vaughan F. R. On the construction of knots and links from Thompson’s groups, Knots, low-dimensional topology and applications (Springer Proc. Math. Stat.), Volume 284, Springer, Cham, 2019, pp. 43-66 https://doi-org.proxy.library.vanderbilt.edu/10.1007/978-3-030-16031-9_3 | DOI | MR | Zbl

[26] Jones, Vaughan F. R. Irreducibility of the Wysiwyg representations of Thompson’s groups, Representation theory, mathematical physics, and integrable systems (Progr. Math.), Volume 340, Birkhäuser/Springer, Cham, 2021, pp. 411-430 | DOI | MR

[27] Mackey, George W. The theory of unitary group representations, University of Chicago Press, Chicago, Ill.-London, 1976, x+372 pages (Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955, Chicago Lectures in Mathematics) | MR | Zbl

[28] Morrison, Scott; Peters, Emiliy; Snyder, Noah Categories generated by a trivalent vertex, Selecta Math. (N.S.), Volume 23 (2017) no. 2, pp. 817-868 https://doi-org.proxy.library.vanderbilt.edu/10.1007/s00029-016-0240-3 | DOI | MR | Zbl

[29] Nikkel, Jordan; Ren, Yunxiang On Jones’ subgroup of R. Thompson’s group T, Internat. J. Algebra Comput., Volume 28 (2018) no. 5, pp. 877-903 https://doi-org.proxy.library.vanderbilt.edu/10.1142/S0218196718500388 | DOI | MR | Zbl

[30] Raghavan, Rushil; Sweeney, Dennis Regular Isotopy Classes of Link Diagrams From Thompson’s Groups (2020) (https://arxiv.org/abs/2008.11052)

[31] Ren, Yunxiang From skein theory to presentations for Thompson group, J. Algebra, Volume 498 (2018), pp. 178-196 https://doi-org.proxy.library.vanderbilt.edu/10.1016/j.jalgebra.2017.11.018 | DOI | MR | Zbl

[32] Savchuk, Dmytro Some graphs related to Thompson’s group F, Combinatorial and geometric group theory (Trends Math.), Birkhäuser/Springer Basel AG, Basel, 2010, pp. 279-296 https://doi-org.proxy.library.vanderbilt.edu/10.1007/978-3-7643-9911-5_12 | DOI | MR

[33] Savchuk, Dmytro Schreier graphs of actions of Thompson’s group F on the unit interval and on the Cantor set, Geom. Dedicata, Volume 175 (2015), pp. 355-372 https://doi-org.proxy.library.vanderbilt.edu/10.1007/s10711-014-9951-9 | DOI | MR | Zbl

Cited by Sources: