These are Lecture Notes of a course given by the author at the School Winter Braids, held at the Université de Pau et des Pays de L’Adour (France), on February 2015. It is explained how mapping class groups, and in particular braid groups, act on some interesting geometric spaces like the hyperbolic plane and the complex of curves, and how this allows to obtain some algebraic properties of the groups. A proof of the hyperbolicity of the graph of curves, following Hensel-Przytycki-Webb, is given.
@article{WBLN_2015__2__A3_0,
author = {Gonz\'alez-Meneses, Juan},
title = {Geometric approaches to braid groups and mapping class groups},
booktitle = {Winter Braids V (Pau, 2015)},
series = {Winter Braids Lecture Notes},
note = {talk:3},
pages = {1--25},
publisher = {Winter Braids School},
year = {2015},
doi = {10.5802/wbln.9},
mrnumber = {3705875},
zbl = {1431.57024},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/wbln.9/}
}
TY - JOUR AU - González-Meneses, Juan TI - Geometric approaches to braid groups and mapping class groups BT - Winter Braids V (Pau, 2015) AU - Collectif T3 - Winter Braids Lecture Notes N1 - talk:3 PY - 2015 SP - 1 EP - 25 PB - Winter Braids School UR - http://archive.numdam.org/articles/10.5802/wbln.9/ DO - 10.5802/wbln.9 LA - en ID - WBLN_2015__2__A3_0 ER -
%0 Journal Article %A González-Meneses, Juan %T Geometric approaches to braid groups and mapping class groups %B Winter Braids V (Pau, 2015) %A Collectif %S Winter Braids Lecture Notes %Z talk:3 %D 2015 %P 1-25 %I Winter Braids School %U http://archive.numdam.org/articles/10.5802/wbln.9/ %R 10.5802/wbln.9 %G en %F WBLN_2015__2__A3_0
González-Meneses, Juan. Geometric approaches to braid groups and mapping class groups, dans Winter Braids V (Pau, 2015), Winter Braids Lecture Notes (2015), Exposé no. 3, 25 p. doi : 10.5802/wbln.9. http://archive.numdam.org/articles/10.5802/wbln.9/
[1] T. Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013), no. 5, 2855–2875. | DOI | MR | Zbl
[2] E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), no. 1, 47-72. | DOI | Zbl
[3] E. Artin, Theory of braids, Ann. of Math. 48 (1947), no. 2, 101-126. | DOI | MR | Zbl
[4] J. S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J. 1974. | DOI
[5] N. Bourbaki, Éléments de mathématique: groupes et algèbres de Lie. Chapitres 4, 5 et 6, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris 1968. | DOI
[6] B. H. Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014), no. 2, 269–280. | DOI | MR | Zbl
[7] M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. | Zbl
[8] M. Calvez, Fast nielsen-thurston classification of braids, Algebr. Geom. Topol. 14 (2014) 1745-1758. | DOI | MR | Zbl
[9] A. Casson, S. A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, 9. Cambridge University Press, Cambridge, 1988. | DOI | Zbl
[10] H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588–621. | DOI | MR | Zbl
[11] A. Clay, K. Rafi, S. Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014), no. 6, 3325–3344. | DOI | MR | Zbl
[12] P. Dehornoy, F. Digne, E. Godelle, D. Krammer, J. Michel, Foundations of Garside Theory, EMS Tracts in Mathematics, volume 22, European Mathematical Society, 2015. | DOI | Zbl
[13] P. Dehornoy, L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3) 79 (1999), no. 3, 569–604. | DOI | MR | Zbl
[14] A. Fathi, F. Laudenbach, V. Poénaru, Travaux de Thurston sur les surfaces, Astérisque No. 66-67 (1991). SMF, Paris, 1991. | Numdam
[15] J. González-Meneses, B. Wiest, On the structure of the centralizer of a braid, Ann. Sci. Norm. Sup. (4) 37 (2004), no. 5, 729–757. | DOI | Numdam | MR | Zbl
[16] J. González-Meneses, The th root of a braid is unique up to conjugacy, Algebr. Geom. Topol. 3 (2003), 1103–1118. | DOI | Zbl
[17] Ursula Hamenstädt, Geometry of the complex of curves and of Teichmüller space, Handbook of Teichmüller theory. Vol. I, IRMA Lect. Math. Theor. Phys., vol. 11, Eur. Math. Soc., Zürich, 2007, 447–467. | DOI | Zbl
[18] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math 84 (1986), no. 1, 157-176. | DOI | MR | Zbl
[19] W. J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 245–251, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981. | DOI
[20] A. Hatcher, On triangulation of surfaces, Topology Appl. 40 (1991), no. 2, 189-194. | DOI | MR | Zbl
[21] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. | Zbl
[22] S. Hensel, P. Przytycki, R. C. Webb, 1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 755–762. | DOI | MR | Zbl
[23] N. V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115. American Mathematical Society, Providence, RI, 1992. | DOI
[24] H. A. Masur, Y. N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. | DOI | MR | Zbl
[25] J. D. McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes, Unpublished. Available at: http://users.math.msu.edu/users/mccarthy/publications/selected.papers.html
[26] Y. Minsky, A geometric approach to the complex of curves, Proceedings of the 37th Taniguchi Symposium on Topology and Teichmüller Spaces (S. Kojima et. al., ed.), World Scientific, 1996, pp. 149–158. | DOI | Zbl
[27] L. Paris, Braid groups and Artin groups, Handbook of Teichmüller theory. Vol. II, 389–451, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009. | DOI | Zbl
[28] K. Rafi, S. Schleimer, Curve complexes are rigid, Duke Math. J. 158 (2011), no. 2, 225-246. | DOI | MR | Zbl
[29] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. | DOI | MR | Zbl
Cité par Sources :
