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, Serie 6, Volume 33 (2024) no. 1, pp. 105-121.

Abstract
Abstract

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 $n \geq 5$ .

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 $n \geq 5$ .

Received: 2020-07-12
Accepted: 2022-02-15
Published online: 2024-07-26

DOI: 10.5802/afst.1763

Author's affiliations:

Stepan Yu. Orevkov ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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

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