Advertisement

Lectures on Artin Groups and the \(K(\pi ,1)\) Conjecture

  • Luis Paris
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 82)

Abstract

This paper consists of the notes of a mini-course (3 lectures) on Artin groups that focuses on a central question of the subject, the \(K(\pi ,1)\) conjecture.

Keywords

Artin group Coxeter group Vinberg system \(K (\pi, 1)\) conjecture 

References

  1. 1.
    Bourbaki, N.: Eléments de mathématique, Fasc. XXXIV: Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris (1968)Google Scholar
  2. 2.
    Brieskorn, E.: Singular elements of semi-simple algebraic groups. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 279–284. Gauthier-Villars, Paris (1971)Google Scholar
  3. 3.
    Brieskorn, E.: Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math. 12, 57–61 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brieskorn, E.: Sur les groupes de tresses [d’après V. I. Arnol’d]. Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401. Lecture Notes in Mathematics, vol. 317, pp. 21–44. Springer, Berlin (1973)Google Scholar
  5. 5.
    Brieskorn, E., Saito, K.: Artin-Gruppen und Coxeter-Gruppen. Invent. Math. 17, 245–271 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Brown, K.S.: Cohomology of Groups: Graduate Texts in Mathematics, 87. Springer-Verlag, New York (1982)Google Scholar
  7. 7.
    Callegaro, F., Moroni, D., Salvetti, M.: The \(K(\pi,1)\) problem for the affine Artin group of type \(\tilde{B}_n\) and its cohomology. J. Eur. Math. Soc. (JEMS) 12(1), 1–22 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Charney, R., Davis, M.W.: Finite \(K(\pi ,1)\)’s for Artin groups. Prospects in Topology (Princeton, NJ, 1994), Annals of Mathematics Studies, 138, pp. 110–124. Princeton University Press, Princeton (1995)Google Scholar
  9. 9.
    Charney, R., Davis, M.W.: The \(K(\pi,1)\)-problem for hyperplane complements associated to infinite reflection groups. J. Amer. Math. Soc. 8(3), 597–627 (1995)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Cox, D., Little, J., O’Shea, D.: Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra (third edition). Undergraduate Texts in Mathematics. Springer, New York (2007)Google Scholar
  11. 11.
    Coxeter, H.S.M.: Discrete groups generated by reflections. Ann. of Math. 35(2), no. 3, 588–621 (1934)Google Scholar
  12. 12.
    Coxeter, H.S.M.: The complete enumeration of finite groups of the form \(R_i^2 = (R_i R_j)^{k_{i, j}} = 1\). J. London Math. Soc. 10, 21–25 (1935)Google Scholar
  13. 13.
    Davis, M.W.: The geometry and topology of Coxeter groups. London Mathematical Society Monographs Series, 32. Princeton University Press, Princeton (2008)Google Scholar
  14. 14.
    Deligne, P.: Les immeubles des groupes de tresses généralisés. Invent. Math. 17, 273–302 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ellis, G., Sköldberg, E.: The \(K(\pi,1)\) conjecture for a class of Artin groups. Comment. Math. Helv. 85(2), 409–415 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Fox, R., Neuwirth, L.: The braid groups. Math. Scand. 10, 119–126 (1962)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Godelle E., Paris, L.: Basic questions on Artin-Tits groups. Configuration Spaces: Geometry, Combinatorics and Topology, pp. 299–311. Edizione della Normale Scuola Normale Superiore, Pisa (2012)Google Scholar
  18. 18.
    Godelle, E., Paris, L.: \(K(\pi,1)\) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups. Math. Z. 272(3), 1339–1364 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  20. 20.
    Hendriks, H.: Hyperplane complements of large type. Invent. Math. 79(2), 375–381 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    van der Lek, H.: The homotopy type of complex hyperplane complements. Ph.D. thesis, University of Nijmegen (1983)Google Scholar
  22. 22.
    Paris, L.: Artin monoids inject in their groups. Comment. Math. Helv. 77(3), 609–637 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Paris, L.: \(K(\pi ,1)\) conjecture for Artin groups. Ann. Fac. Sci. Toulouse Math. (to appear)Google Scholar
  24. 24.
    Tits, J.: Groupes et géométries de Coxeter. Institut des Hautes Etudes Scientifiques, Paris (1961)Google Scholar
  25. 25.
    Tits, J.: Normalisateurs de tores. I. Groupes de Coxeter étendus. J. Algebra 4, 96–116 (1966)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Tits, J.: Le problème des mots dans les groupes de Coxeter. Symposia Mathematica (INDAM, Rome, 1967/68), vol. 1, pp. 175–185. Academic Press, London (1969)Google Scholar
  27. 27.
    Vinberg, È.B.: Discrete linear groups that are generated by reflections. Izv. Akad. Nauk SSSR Ser. Mat. 35, 1072–1112 (1971)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer India 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de BourgogneUniversité de BourgogneDijon cedexFrance

Personalised recommendations