Coxeter-type quotients of surface braid groups

Renato Diniz; Oscar Ocampo; Paulo Cesar Cerqueira dos Santos Júnior

doi:10.5802/crmath.813

Article de recherche - Géométrie et Topologie

Coxeter-type quotients of surface braid groups
[Quotients de type Coxeter des groupes de tresses sur les surfaces]

Renato Diniz ¹ ; Oscar Ocampo ² ; Paulo Cesar Cerqueira dos Santos Júnior ³

¹Universidade Federal do Recôncavo da Bahia – CFP, Av. Nestor de Melo Pita, 535, CEP: 45300.000, Amargosa, BA, Brazil
²Universidade Federal da Bahia, Departamento de Matemática – IME, Av. Milton Santos S/N, CEP: 40170-110, Salvador, BA, Brazil
³Universidade Estadual do Sudoeste da Bahia, Departamento de Ciências Exatas e Tecnológicas – DCET, Estrada do Bem Querer S/N, CEP: 45031-900, Vitória da Conquista, BA, Brazil

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 27-37

Abstract (VO)
Résumé

Let $M$ be a closed surface, $q\ge 2$ and $n\ge 2$. In this paper, we analyze the Coxeter-type quotient group $B_n(M)(q)$ of the surface braid group $B_{n}(M)$ by the normal closure of the element $\sigma _1^q$, where $\sigma _1$ is the standard Artin generator of the braid group $B_n$. Also, we study the Coxeter-type quotient groups obtained by taking the quotient of $B_n(M)$ by the commutator subgroup of the respective pure braid group $\bigl [{P_n(M),P_n(M)}\bigr ]$ and adding the relation $\sigma _1^q=1$, when $M$ is a closed orientable surface or the disk.

Soit $M$ une surface fermée, $q\ge 2$ et $n\ge 2$. Dans cet article, nous étudions le groupe quotient de type Coxeter $B_n(M)(q)$ du groupe de tresses sur la surface $B_n(M)$, défini comme le quotient par la clôture normale de l’élément $\sigma _1^q$, où $\sigma _1$ désigne le générateur d’Artin standard du groupe de tresses $B_n$. Nous étudions également les groupes quotients de type Coxeter obtenus en quotientant $B_n(M)$ par le sous-groupe dérivé du groupe de tresses pures correspondant $\bigl [{P_n(M),P_n(M)}\bigr ]$ et en ajoutant la relation $\sigma _1^q=1$, lorsque $M$ est une surface orientable fermée ou le disque.

Reçu le : 2024-12-19
Révisé le : 2025-12-18
Accepté le : 2025-12-18
Publié le : 2026-02-16

DOI : 10.5802/crmath.813

Classification : 20F36, 20F05
Keywords: Artin braid group, surface braid group, finite group
Mots-clés : Groupe de tresses d’Artin, groupe de tresses sur une surface, groupe fini

Affiliations des auteurs :

Renato Diniz ¹ ; Oscar Ocampo ² ; Paulo Cesar Cerqueira dos Santos Júnior ³

¹ Universidade Federal do Recôncavo da Bahia – CFP, Av. Nestor de Melo Pita, 535, CEP: 45300.000, Amargosa, BA, Brazil
² Universidade Federal da Bahia, Departamento de Matemática – IME, Av. Milton Santos S/N, CEP: 40170-110, Salvador, BA, Brazil
³ Universidade Estadual do Sudoeste da Bahia, Departamento de Ciências Exatas e Tecnológicas – DCET, Estrada do Bem Querer S/N, CEP: 45031-900, Vitória da Conquista, BA, Brazil

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2026__364_G1_27_0,
     author = {Renato Diniz and Oscar Ocampo and Paulo Cesar Cerqueira dos Santos J\'unior},
     title = {Coxeter-type quotients of surface braid groups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {27--37},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.813},
     language = {en},
}

TY  - JOUR
AU  - Renato Diniz
AU  - Oscar Ocampo
AU  - Paulo Cesar Cerqueira dos Santos Júnior
TI  - Coxeter-type quotients of surface braid groups
JO  - Comptes Rendus. Mathématique
PY  - 2026
SP  - 27
EP  - 37
VL  - 364
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.813
LA  - en
ID  - CRMATH_2026__364_G1_27_0
ER  -

%0 Journal Article
%A Renato Diniz
%A Oscar Ocampo
%A Paulo Cesar Cerqueira dos Santos Júnior
%T Coxeter-type quotients of surface braid groups
%J Comptes Rendus. Mathématique
%D 2026
%P 27-37
%V 364
%I Académie des sciences, Paris
%R 10.5802/crmath.813
%G en
%F CRMATH_2026__364_G1_27_0

Renato Diniz; Oscar Ocampo; Paulo Cesar Cerqueira dos Santos Júnior. Coxeter-type quotients of surface braid groups. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 27-37. doi: 10.5802/crmath.813

Bibliographie
Cité par

[1] Noneuclidean tesselations and their groups (Wilhelm Magnus, ed.), Pure and Applied Mathematics, Academic Press Inc., 1974 | Zbl | DOI | MR

[2] E. Artin Theorie der Zöpfe, Abh. Math. Semin. Univ. Hamb., Volume 4 (1925), pp. 47-72 | DOI | Zbl | MR

[3] E. Artin Theory of braids, Ann. Math. (2), Volume 48 (1947), pp. 101-126 | DOI | MR | Zbl

[4] Joachim Assion A proof of a theorem of Coxeter, C. R. Math. Rep. Acad. Sci. Canada, Volume 1 (1978/79) no. 1, pp. 41-44 | MR | Zbl

[5] Ishan Banerjee; Peter Huxford Generators for the level $m$ congruence subgroups of braid groups, J. Algebra, Volume 691 (2026), pp. 1-16 | DOI | MR | Zbl

[6] Paolo Bellingeri; Celeste Damiani; Oscar Ocampo; Charalampos Stylianakis Powers of half-twists and congruence subgroups of braid groups (2025) | arXiv

[7] James van Buskirk Braid groups of compact $2$ -manifolds with elements of finite order, Trans. Am. Math. Soc., Volume 122 (1966), pp. 81-97 | DOI | MR | Zbl

[8] H. S. M. Coxeter Factor groups of the braid group, Proceedings of the Fourth Canadian Mathematical Congress (Banff, 1957) (M. S. MacPhail, ed.), University of Toronto Press, 1959, pp. 95-122 | Zbl

[9] Edward Fadell; James Van Buskirk The braid groups of $E^{2}$ and $S^{2}$ , Duke Math. J., Volume 29 (1962), pp. 243-257 | MR | Zbl

[10] R. Fox; L. Neuwirth The braid groups, Math. Scand., Volume 10 (1962), pp. 119-126 | DOI | MR | Zbl

[11] Katherine M. Goldman $CAT (0)$ and cubulated Shephard groups, J. Lond. Math. Soc. (2), Volume 111 (2025) no. 1, e70050, 33 pages | DOI | MR | Zbl

[12] Daciberg Lima Gonçalves; John Guaschi; Oscar Ocampo A quotient of the Artin braid groups related to crystallographic groups, J. Algebra, Volume 474 (2017), pp. 393-423 | DOI | MR | Zbl

[13] Daciberg Lima Gonçalves; John Guaschi; Oscar Ocampo; Carolina de Miranda e Pereiro Crystallographic groups and flat manifolds from surface braid groups, Topology Appl., Volume 293 (2021), 107560, 16 pages | DOI | MR | Zbl

[14] Juan González-Meneses New presentations of surface braid groups, J. Knot Theory Ramifications, Volume 10 (2001) no. 3, pp. 431-451 | DOI | MR | Zbl

[15] Juan González-Meneses; Luis Paris Vassiliev invariants for braids on surfaces, Trans. Am. Math. Soc., Volume 356 (2004) no. 1, pp. 219-243 | DOI | MR | Zbl

[16] John Guaschi; Daniel Juan-Pineda A survey of surface braid groups and the lower algebraic $K$ -theory of their group rings, Handbook of group actions. Vol. II (Lizhen Ji; Athanase Papadopoulos; Shing-Tung Yau, eds.) (Advanced Lectures in Mathematics), Volume 32, International Press, 2015, pp. 23-75 | MR

[17] Vagn Lundsgaard Hansen Braids and coverings: selected topics, London Mathematical Society Student Texts, 18, Cambridge University Press, 1989 | DOI | MR | Zbl

[18] Gareth A. Jones; J. Mary Jones Elementary number theory, Springer Undergraduate Mathematics Series, Springer, 1998 | DOI | MR | Zbl

[19] Kunio Murasugi; Bohdan I. Kurpita A study of braids, Mathematics and its Applications, 484, Kluwer Academic Publishers, 1999 | DOI | MR | Zbl

[20] Paulo Cesar Cerqueira dos Santos Júnior Um quociente do grupo de tranças de Artin relacionado aos grupos cristalográficos, Ph. D. Thesis, Universidade Federal da Bahia, Salvador (Brazil) (2019)

[21] G. P. Scott Braid groups and the group of homeomorphisms of a surface, Proc. Camb. Philos. Soc., Volume 68 (1970), pp. 605-617 | DOI | MR | Zbl

[22] The GAP Group GAP – Groups, Algorithms, and Programming, version 4.12.2, 2022 https://www.gap-system.org

[23] Bronislaw Wajnryb A braidlike presentation of $Sp (n, p)$ , Isr. J. Math., Volume 76 (1991) no. 3, pp. 265-288 | DOI | MR | Zbl

[24] Oscar Zariski The topological discriminant group of a Riemann surface of genus $p$ , Am. J. Math., Volume 59 (1937) no. 2, pp. 335-358 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique