Seminar: Coxeter Groups

A reflection group (or more general, a Coxeter group) is the group of symmetries of a "geometric object" generated by a set of reflections; for example, a group of symmetries of the pentagonal tiling of the hyperbolic plane generated by the reflections along the edges of a given pentagon. Coxeter groups exhibit interesting algebraic and topological properties and provide a great playground to apply methods from geometry group theory, algebraic topology, and algebra.

reflection group

Coxeter groups play an important role in several areas of mathematics. Among others, they were used to construct various examples of groups and manifolds with unexpected properties. E.g., to construct exotic examples of aspherical manifolds (an aspherical manifold is a manifold whose universal cover is contractible).

The purpose of this seminar is to introduce Coxeter groups, study their basic properties and examples coming from geometry (as tilings and polygons). Then, we introduce the main geometric object assigned to each Coxeter group W, called the Davis' complex. This complex is a natural space for which W is the group of symmetries. Thus it generalises, e.g., the concept of a tiling of hyperbolic or Euclidean space. The important feature which the Davis complex shares with Euclidean or hyperbolic space is that it is "non-positively curved" and contractible. As the last goal, we want to discuss some constructions of exotic aspherical manifolds. To attend the seminar, a basic knowledge of algebraic topology (fundamental group, homology theory) is expected. Familiarity with basic concepts of geometric group theory will be helpful but is not required.

The main reference for the seminar will be the book The Geometry and Topology of Coxeter Groups by M. Davis.

This seminar will be held in English. The written report can be in English or German. The handouts should be in English.

N. Bourbaki. Groupes et algèbres de Lie. Chapitre IV, V, VI.
M. R. Bridson, A. Haefliger. Metric spaces of non-positive curvature.
M. W. Davis and G. Moussong. Notes on nonpositively curved polyhedra.
M. W. Davis. The geometry and topology of Coxeter groups.
M. W. Davis. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2), 117(2):293--324, 1983.