Seminar: Coxeter Groups, SS 2016
Prof. Dr. C. Löh
Dr. Michał Marcinkowski
The schedule is now available. There are still free slots!
The organisational meeting for this seminar will be on Wednesday, January 27, at 9:15
(M 101). Alternatively, you can register for this seminar by sending an email
to firstname.lastname@example.org .
If you have any questions about the seminar or your talk, you can ask
Michał Marcinkowski (email@example.com, M 205) for assistance.
This seminar will be held in English. The written report
can be in English or German. The handouts should be in English.
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 in the edges of a
given pentagon. Coxeter groups exhibit interesting algebraic and
topological properties and provide a great playground to apply methods
from geometric group theory, algebraic topology, and algebra.
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.
Wednesdays, 8:30 -- 10:00
Hints on writing mathematical texts (in German)
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.
To attend the seminar, a basic knowledge of algebraic topology
(fundamental group, homology theory) is expected
(as, e.g., covered in the course
Algebraic Topology I
in WS 2015/16).
Familiarity with basic concepts of geometric group theory will be
helpful but is not required.
Sufficient for successfully passing this seminar are:
Giving a presentation.
Active participation in the seminar sessions.
A handout (one or two pages), containing a summary of the central
aspects of the talk as well as some exercises.
A detailed written report (in German or English) on the topic of the talk
(to be handed in one week before the talk).
Depending on the applicable Prüfungsordnung/Modulkatalog, the grade
will be based either only on the presentation or only on the written report.
Depending on the applicable Prüfungsordnung/Modulkatalog, this
seminar will be awarded with 6 or 4.5 credits.
This seminar is suitable for Bachelor/Master students and
Lehramtsstudenten with an interest in topology, and could be the
starting point for a project under my supervision (e.g., bachelor
or master thesis).
Last Change: January 12, 2016.