logo AFST
Automorphism group of the commutator subgroup of the braid group
Stepan Yu. Orevkov
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, p. 1137-1161

Let B n ' be the commutator subgroup of the braid group B n . We prove that Aut(B n ' )=Aut(B n ) for n4. This answers a question asked by Vladimir Lin.

Soit B n ' le groupe dérivé du groupe de tresses B n . On montre que Aut(B n ' )=Aut(B n ) pour n4, ce qui répond à une question posée par Vladimir Lin.

Received : 2015-11-11
Accepted : 2016-07-05
Published online : 2017-12-15
DOI : https://doi.org/10.5802/afst.1562
@article{AFST_2017_6_26_5_1137_0,
     author = {Stepan Yu. Orevkov},
     title = {Automorphism group of the commutator subgroup of the braid group},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {5},
     year = {2017},
     pages = {1137-1161},
     doi = {10.5802/afst.1562},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_5_1137_0}
}
Orevkov, Stepan Yu. Automorphism group of the commutator subgroup of the braid group. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 5, pp. 1137-1161. doi : 10.5802/afst.1562. afst.centre-mersenne.org/item/AFST_2017_6_26_5_1137_0/

[1] Emil Artin Theory of braids, Ann. Math., Tome 48 (1946), pp. 101-126 | Article

[2] Joan Birman; Ki Hyoung Ko; Sang Jin Lee A new approach to the word and conjugacy problems in the braid groups, Adv. Math., Tome 139 (1998) no. 2, pp. 322-353 | Article

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

[4] John Crisp; Luis Paris The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math., Tome 145 (2001) no. 1, pp. 19-36 | Article

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

[6] Samuel Eilenberg Sur les transformations périodiques de la surface de sphère, Fundamenta Math., Tome 22 (1934), pp. 28-41 | Article

[7] William Fulton; Joe Harris Representation theory. A first course, Graduate Texts in Mathematics, Tome 129, Springer, 1991, xv+551 pages

[8] Volker Gebhardt; Juan González-Meneses The cyclic sliding operation in Garside groups, Math. Z., Tome 265 (2010) no. 1, pp. 85-114 | Article

[9] Juan González-Meneses The nth root of a braid is unique up conjugacy, Algebr. Geom. Topol., Tome 3 (2003), pp. 1103-1118 | Article

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

[11] Evgeny A. Gorin; Vladimir Lin Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR-Sbornik, Tome 7 (1969), pp. 569-596 | Article

[12] Nikolaj V. Ivanov Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, Tome 115, American Mathematical Society, 1992, xii+127 pages

[13] Béla von Kerékjártó Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann., Tome 80 (1919), pp. 36-38 | Article

[14] Christopher J. Leininger; Dan Margalit Two-generator subgroups of the pure braid group, Geom. Dedicata, Tome 147 (2010), pp. 107-113 | Article

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

[16] Vladimir 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)

[17] Kay Magaard; Gunter Malle; Pham Huu Tiep Irreducibility of tensor squares, symmetric squares and alternating squares, Pac. J. Math., Tome 202 (2002) no. 2, pp. 379-427 | Article

[18] Wilhelm Magnus; Abraham Karrass; Donald Solitar Combinatorial group theory: presentations of groups in terms of generators and relations, Pure and Applied Mathematics, Tome 13, Interscience Publishers, 1966, xii+444 pages

[19] Sandro Manfredini Some subgroups of Artin’s braid group, Topology Appl., Tome 78 (1997) no. 1-2, pp. 123-142 | Article

[20] Stepan Yu. Orevkov Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves, J. Knot Theory Ramifications, Tome 10 (2001) no. 7, pp. 1005-1023 | Article

[21] Stepan Yu. Orevkov Algorithmic recognition of quasipositive braids of algebraic length two, J. Algebra, Tome 423 (2015), pp. 1080-1180 | Article

[22] Stepan Yu. Orevkov Algorithmic recognition of quasipositive 4-braids of algebraic length three, Groups Complex. Cryptol., Tome 7 (2015) no. 2, pp. 157-173 | Article