×

zbMATH — the first resource for mathematics

On the structure of the centralizer of a braid. (English) Zbl 1082.20024
For any given element of the \(n\)-string braid group, a small generating set of its centralizer is computed. This leads to sharp bounds for the size of a generating set for the centalizer, which is described in terms of semidirect and direct products of mixed braid groups. Nice figures and examples accompany the theorems. An explicit construction for the generating set is given.

MSC:
20F36 Braid groups; Artin groups
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
57M25 Knots and links in the \(3\)-sphere (MSC2010)
Software:
Trains
PDF BibTeX XML Cite
Full Text: DOI arXiv Numdam EuDML Link
References:
[1] Arnold V.I. , The cohomology ring of the group of dyed braids , Mat. Zametki 5 ( 1969 ) 227 - 231 . MR 242196 | Zbl 0277.55002 · Zbl 0277.55002
[2] Benardete D. , Gutierrez M. , Nitecki Z. , A combinatorial approach to reducibility of mapping classes , Contemporary Math. 150 ( 1993 ) 1 - 31 . MR 1234257 | Zbl 0804.57005 · Zbl 0804.57005
[3] Benardete D. , Gutierrez M. , Nitecki Z. , Braids and the Nielsen-Thurston classification , J. Knot Theory Ramifications 4 ( 1995 ) 549 - 618 . MR 1361083 | Zbl 0874.57010 · Zbl 0874.57010
[4] Bessis D. , Digne F. , Michel J. , Springer theory in braid groups and the Birman-Ko-Lee monoid , Pacific J. Math. 205 ( 2 ) ( 2002 ) 287 - 309 . MR 1922736 | Zbl 1056.20023 · Zbl 1056.20023
[5] Bestvina M. , Haendel M. , Train-tracks for surface homeomorphisms , Topology 34 ( 1995 ) 109 - 140 . MR 1308491 | Zbl 0837.57010 · Zbl 0837.57010
[6] Birman J. , Braids, Links, and Mapping Class Groups , Annals of Math. Studies , vol. 82 , Princeton University Press , 1975 . MR 375281 | Zbl 0305.57013 · Zbl 0305.57013
[7] Birman J. , Ko K.H. , Lee S.J. , A new approach to the word and conjugacy problems in the braid groups , Adv. Math. 139 ( 1998 ) 322 - 353 . MR 1654165 | Zbl 0937.20016 · Zbl 0937.20016
[8] Birman J. , Lubotzky A. , McCarthy J. , Abelian and solvable subgroups of the mapping class groups , Duke Math. J. 50 ( 1983 ) 1107 - 1120 . Article | MR 726319 | Zbl 0551.57004 · Zbl 0551.57004
[9] Brinkmann P. , An implementation of the Bestvina-Handel algorithm for surface homeomorphisms , Experiment. Math. 9 ( 2000 ) 235 - 240 , Computer program available at, http://www.math.uiuc.edu/ brinkman/software/train/ . Article | MR 1780208 | Zbl 0982.57005 · Zbl 0982.57005
[10] Burde G. , Über Normalisatoren der Zopfgruppe , Abh. Math. Sem. Univ. Hamburg 27 ( 1964 ) 97 - 115 . MR 170954 | Zbl 0134.43104 · Zbl 0134.43104
[11] Constantin A. , Kolev B. , The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere , Enseign. Math. (2) 40 ( 3-4 ) ( 1994 ) 193 - 204 . MR 1309126 | Zbl 0852.57012 · Zbl 0852.57012
[12] Eilenberg S. , Sur les transformations périodiques de la surface de sphère , Fund. Math. 22 ( 1934 ) 28 - 41 . Article | Zbl 0008.37109 · Zbl 0008.37109
[13] El-Rifai E.A. , Morton H.R. , Algorithms for positive braids , Quart. J. Math. Oxford Ser. (2) 45 ( 180 ) ( 1994 ) 479 - 497 . MR 1315459 | Zbl 0839.20051 · Zbl 0839.20051
[14] Fathi A. , Laudenbach F. , Poenaru V. , Travaux de Thurston sur les surfaces - séminaire Orsay , Astérisque , vols. 66-67 , Société Math. de France , 1991 . MR 1134426 | Zbl 0406.00016 · Zbl 0406.00016
[15] Fenn R. , Rolfsen D. , Zhu J. , Centralisers in the braid group and singular braid monoid , Enseign. Math. 42 ( 1996 ) 75 - 96 . MR 1395042 | Zbl 0869.20024 · Zbl 0869.20024
[16] Franco N. , González-Meneses J. , Conjugacy problem for braid groups and Garside groups , J. Algebra 266 ( 2003 ) 112 - 132 . MR 1994532 | Zbl 1043.20019 · Zbl 1043.20019
[17] Franco N. , González-Meneses J. , Computation of centralizers in braid groups and Garside groups , Rev. Mat. Iberoamericana 19 ( 2003 ) 367 - 384 . Article | MR 2023190 | Zbl 1064.20040 · Zbl 1064.20040
[18] Garside F.A. , The braid group and other groups , Quart. J. Math. Oxford 20 ( 1969 ) 235 - 254 . MR 248801 | Zbl 0194.03303 · Zbl 0194.03303
[19] Gebhardt V. , A new approach to the conjugacy problem in Garside groups , Preprint, math.GT/0306199 . arXiv | MR 2166805 · Zbl 1105.20032
[20] Hall T. , Computer implementation of Bestvina-Handel algorithm , available at, http://www.liv.ac.uk/maths/PURE/MIN_SET/CONTENT/members/T_Hall.html .
[21] Ivanov N.V. , Subgroups of Teichmüller Modular Groups , Translations of Mathematical Monographs , vol. 115 , AMS , 1992 . MR 1195787 | Zbl 0776.57001 · Zbl 0776.57001
[22] Ivanov N.V., Talk at the special session “Mapping class groups and the geometric theory of Teichmüller spaces” at the 974th meeting of the AMS, Ann Harbour, MI, March 1-3, 2002.
[23] Ivanov N.V. , Examples of centralizers in the Artin braid groups , Preprint, math.GT/0306418 . arXiv
[24] de Kerékjártó B. , Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche , Math. Annalen 80 ( 1919 ) 3 - 7 . JFM 47.0526.05 · JFM 47.0526.05
[25] Los J. , Pseudo-Anosov maps and invariant train tracks in the disc: a finite algorithm , Proc. London Math. Soc. (3) 66 ( 1993 ) 400 - 430 . MR 1199073 | Zbl 0788.58039 · Zbl 0788.58039
[26] Makanin G.S. , On normalizers in the braid group , Mat. Sb. 86 ( 128 ) ( 1971 ) 171 - 179 . MR 347988 | Zbl 0229.20035 · Zbl 0229.20035
[27] Manfredini S. , Some subgroups of Artin’s braid group. Special issue on braid groups and related topics (Jerusalem, 1995) , Topology Appl. 78 ( 1-2 ) ( 1997 ) 123 - 142 . MR 1465028 | Zbl 0965.20016 · Zbl 0965.20016
[28] Orevkov S.Yu. , Quasipositivity test via unitary representations of braid groups and its applications to real algebraic curves , J. Knot Theory Ramifications 10 ( 7 ) ( 2001 ) 1005 - 1023 . MR 1867106 | Zbl 1030.20026 · Zbl 1030.20026
[29] Paris L. , Rolfsen D. , Geometric subgroups of surface braid groups , Ann. Inst. Fourier 49 ( 1999 ) 101 - 156 . Numdam | MR 1697370 | Zbl 0962.20028 · Zbl 0962.20028
[30] Penner R.C. , Harer J.L. , Combinatorics of Train Tracks , Annals of Math. , vol. 125 , Princeton University Press , Princeton, NJ , 1992 . MR 1144770 | Zbl 0765.57001 · Zbl 0765.57001
[31] Sibert H. , Extraction of roots in Garside groups , Comm. Algebra 30 ( 6 ) ( 2002 ) 2915 - 2927 . MR 1908246 | Zbl 1007.20036 · Zbl 1007.20036
[32] Styšnev V.B. , Izv. Akad. Nauk SSSR Ser. Mat. 42 ( 5 ) ( 1978 ) 1120 - 1131 , 1183. MR 513916
[33] Thurston W.P. , On the geometry and dynamics of diffeomorphisms of surfaces , Bull. Amer. Math. Soc. (N.S.) 19 ( 1988 ) 417 - 431 . Article | MR 956596 | Zbl 0674.57008 · Zbl 0674.57008
[34] Thurston W.P. , Braid Groups , in: Epstein D.B.A. , Cannon J.W. , Holt D.F. , Levy S.V.F. , Paterson M.S. , Thurston W.P. (Eds.), Word Processing in Groups , Jones and Bartlett Publishers , Boston, MA , 1992 , Chapter 9. MR 1161694 | Zbl 0764.20017 · Zbl 0764.20017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.