The Magnus representation of the Torelli group $\mathcal {I}_{g,1}$ is not faithful for $g \geq 2$
HTML articles powered by AMS MathViewer
- by Masaaki Suzuki PDF
- Proc. Amer. Math. Soc. 130 (2002), 909-914 Request permission
Abstract:
In this paper we consider the Magnus representation of the Torelli group. We prove that it is not faithful by showing a non-trivial element in the kernel of this representaion.References
- S. Bigelow The Burau representation is not faithful for $n=5$, Geom. Topol. 3 (1999), 397–404
- J. Birman, Braids, Links and Mapping Class Groups, Ann. of Math. Stud. No. 82, Princeton University Press, 1975
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Dennis Johnson, A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981) Contemp. Math., vol. 20, Amer. Math. Soc., Providence, RI, 1983, pp. 165–179. MR 718141, DOI 10.1090/conm/020/718141
- D. D. Long and M. Paton, The Burau representation is not faithful for $n\geq 6$, Topology 32 (1993), no. 2, 439–447. MR 1217079, DOI 10.1016/0040-9383(93)90030-Y
- John Atwell Moody, The Burau representation of the braid group $B_n$ is unfaithful for large $n$, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 379–384. MR 1098347, DOI 10.1090/S0273-0979-1991-16080-5
- Jürgen Aldinger, Isaac Klapper, and Michael Tabor, Formulae for the calculation and estimation of writhe, J. Knot Theory Ramifications 4 (1995), no. 3, 343–372. MR 1347359, DOI 10.1142/S021821659500017X
- Shigeyuki Morita, The extension of Johnson’s homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993), no. 1, 197–224. MR 1193604, DOI 10.1007/BF01231286
- Shigeyuki Morita, Problems on the structure of the mapping class group of surfaces and the topology of the moduli space of curves, Topology, geometry and field theory, World Sci. Publ., River Edge, NJ, 1994, pp. 101–110. MR 1312176
- Shigeyuki Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 349–406. MR 1734418, DOI 10.2140/gtm.1999.2.349
Additional Information
- Masaaki Suzuki
- Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan
- Email: macky@ms.u-tokyo.ac.jp
- Received by editor(s): May 16, 2000
- Received by editor(s) in revised form: August 30, 2000
- Published electronically: June 8, 2001
- Communicated by: Ronald A. Fintushel
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 909-914
- MSC (2000): Primary 57M05; Secondary 20F34, 57N05
- DOI: https://doi.org/10.1090/S0002-9939-01-06128-7
- MathSciNet review: 1866048