The CAT(0) geometry of Coxeter groups and Artin groups 2022

Instructor: Alexander Goldman.

Syllabus: Introduction to CAT(0) spaces and complexes of groups using Coxeter groups and Artin groups as the primary motivation and examples. Material will jump between the following topics:

  1. CAT(κ) spaces: the model spaces Mκ, CAT(κ) polyhedral complexes (including CAT(0) cube complexes), the link condition.
  2. Complexes of groups: simple complexes of groups, the local development, global developability, the universal and fundamental groups, metrics of nonpositive curvature.
  3. Coxeter groups: spherical (finite) linear reflection groups, the Davis-Moussong complex Σ for infinite Coxeter groups, the CAT(0) metric on Σ.
  4. Artin groups: hyperplane arrangements, spherical-type Artin groups, the Deligne complex Φ for infinite-type Artin groups, right-angled Artin groups (including special cube complexes), type FC Artin groups and metrics on Φ, the K(π,1) conjecture.

Schedule: We will have three lectures per week, on Mondays, Wednesdays, and Fridays, from 2 p.m. to 3:30 p.m. 3 p.m., starting on May 30th, and ending on June 17th. Online on Zoom.

Main reference: Lecture notes written by Alexander specifically for this course.


Book references:

  • Bridson, M. R.; Haefliger, A.; Metric spaces of non-positive curvature. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. xxii+643 pp. ISBN: 3-540-4324-9.
  • Bourbaki, N.; Lie groups and Lie algebras. Chapters 4–6. Translated from the 1968 French original by Andrew Pressley. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. xii+300 pp. ISBN: 3-540-42650-7.
  • Davis, M. W.; The geometry and topology of Coxeter groups. London Mathematical Society Monographs Series, 32. Princeton University Press, Princeton, NJ, 2008. xvi+584 pp. ISBN: 978-0-691-13138-2; 0-691-13138-4.

Article references:

  • Deligne, P.; Les immeubles des groupes de tresses généralisés. (French) Invent. Math. 17 (1972), 273–302.
  • Haefliger, A.; Extension of complexes of groups. (French) Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 275–311.
  • Charney, R.; Davis, M. W.; The K(π,1)-problem for hyperplane complements associated to infinite reflection groups. J. Amer. Math. Soc. 8 (1995), no. 3, 597–627.