Homomorphisms of commutator subgroups of braid groups with small number of strings

Stepan Yu. Orevkov

doi:10.5802/afst.1763

Stepan Yu. Orevkov ¹

¹ IMT, Univ. Paul Sabatier, Toulouse, France — Steklov Math. Inst., Moscow, Russia

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 105-121.

Résumé
Abstract

Pour tout $n$ nous décrivons tous les endomophismes du groupe de tresses $B_{n}$ et de son sous-groupe dérivé $B_{n}^{'}$ ainsi que tous les homomorphismes $B_{n}^{'} \to B_{n}$ . Ces résultats ne sont nouveaux que pour $n$ petits parce que les endomorphismes de $B_{n}$ sont déjà décrits par Castel pour $n \geq 6$ et les homomorphismes $B_{n}^{'} \to B_{n}$ ainsi que les endomorphismes de $B_{n}^{'}$ sont décrits par Kordek et Margalit pour $n \geq 7$ . Nous utilisons des approches très différentes pour $n = 4$ et pour $n \geq 5$ .

For any $n$ , we describe all endomorphisms of the braid group $B_{n}$ and of its commutator subgroup $B_{n}^{'}$ , as well as all homomorphisms $B_{n}^{'} \to B_{n}$ . These results are new only for small $n$ because endomorphisms of $B_{n}$ are already described by Castel for $n \geq 6$ , and homomorphisms $B_{n}^{'} \to B_{n}$ and endomorphisms of $B_{n}^{'}$ are already described by Kordek and Margalit for $n \geq 7$ . We use very different approaches for $n = 4$ and for $n \geq 5$ .

Reçu le : 2020-07-12
Accepté le : 2022-02-15
Publié le : 2024-07-26

DOI : 10.5802/afst.1763

Affiliations des auteurs :

Stepan Yu. Orevkov ¹

¹ IMT, Univ. Paul Sabatier, Toulouse, France — Steklov Math. Inst., Moscow, Russia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AFST_2024_6_33_1_105_0,
     author = {Stepan Yu. Orevkov},
     title = {Homomorphisms of commutator subgroups of braid groups with small number of strings},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {105--121},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 33},
     number = {1},
     year = {2024},
     doi = {10.5802/afst.1763},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1763/}
}

TY  - JOUR
AU  - Stepan Yu. Orevkov
TI  - Homomorphisms of commutator subgroups of braid groups with small number of strings
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2024
SP  - 105
EP  - 121
VL  - 33
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1763/
DO  - 10.5802/afst.1763
LA  - en
ID  - AFST_2024_6_33_1_105_0
ER  -

%0 Journal Article
%A Stepan Yu. Orevkov
%T Homomorphisms of commutator subgroups of braid groups with small number of strings
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2024
%P 105-121
%V 33
%N 1
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1763/
%R 10.5802/afst.1763
%G en
%F AFST_2024_6_33_1_105_0

Stepan Yu. Orevkov. Homomorphisms of commutator subgroups of braid groups with small number of strings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 1, pp. 105-121. doi : 10.5802/afst.1763. https://afst.centre-mersenne.org/articles/10.5802/afst.1763/

Bibliographie
Cité par

[1] Emil Artin Theory of braids, Ann. Math., Volume 48 (1947), pp. 101-126 | DOI | Zbl

[2] Robert W. Bell; Dan Margalit Braid groups and the co-Hopfian property, J. Algebra, Volume 303 (2006) no. 1, pp. 275-294 | DOI | Zbl

[3] Joan S. Birman; Alex Lubotzky; John McCarthy Abelian and solvable subgroups of the mapping class group, Duke Math. J., Volume 50 (1983), pp. 1107-1120 | Zbl

[4] Fabrice Castel Geometric representations of the braid groups, Astérisque, 378, Société Mathématique de France, 2016, vi+175 pages | Zbl

[5] Joan L. Dyer; Edna K. Grossman The automorphism group of the braid groups, Am. J. Math., Volume 103 (1981), pp. 1151-1169 | DOI | Zbl

[6] Benson Farb; Dan Margalit A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012 | Zbl

[7] Juan Gonzalez-Meneses The $n$ th root of a braid is unique up conjugacy, Algebr. Geom. Topol., Volume 3 (2003), pp. 1103-1118 | DOI | Zbl

[8] Juan González-Meneses; Bert Wiest On the structure of the centralizer of a braid, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 5, pp. 729-757 | DOI | Numdam | Zbl

[9] Evgeny A. Gorin; Vladimir Ya. Lin Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR, Sb., Volume 7 (1969), pp. 569-596 | DOI | Zbl

[10] Nikolaj V. Ivanov Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115, American Mathematical Society, 1992, xii+127 pages | DOI | Zbl

[11] Kevin Kordek; Dan Margalit Homomorphisms of commutator subgroups of braid groups, Bull. Lond. Math. Soc., Volume 54 (2022) no. 1, pp. 95-111 | DOI | Zbl

[12] Vladimir Ya. Lin Braids and Permutations (2004) (https://arxiv.org/abs/math/0404528)

[13] Vladimir Ya. Lin Algebraic functions, configuration spaces, Teichmüller spaces, and new holomorphically combinatorial invariants, Funkts. Anal. Prilozh., Volume 45 (2011) no. 3, pp. 55-78 English transl. in Funct. Anal. Appl 45 (2011), no. 3, p. 204–224 | Zbl

[14] Vladimir Ya. Lin Some problems that I would like to see solved, 2015 (Abstract of a talk. Technion. http://www2.math.technion.ac.il/~pincho/Lin/Abstracts.pdf)

[15] Wilhelm Magnus; Abraham Karrass; Donald Solitar Combinatorial group theory: presentations of groups in terms of generators and relations, Pure and Applied Mathematics, 13, Interscience Publishers, 1966 | Zbl

[16] Stepan Yu. Orevkov Algorithmic recognition of quasipositive braids of algebraic length two, J. Algebra, Volume 423 (2015), pp. 1080-1108 | DOI | Zbl

[17] Stepan Yu. Orevkov Automorphism group of the commutator subgroup of the braid group, Ann. Fac. Sci. Toulouse, Math., Volume 26 (2017) no. 5, pp. 1137-1161 | DOI | Numdam | Zbl

Cité par Sources :