Seminar: The Geometry and Topology of Coxeter Groups

Sommersemester 21

Dr. Bakul Sathaye and Prof. Linus Kramer

This is a seminar course aimed at Masters students with some background in group theory and geometry. This would be well suited for students who took the course on “Nonpositively curved spaces” in Winter 2020/21, or a course on Geometric Group Theory. The seminar will also be suitable for Bachelors students which have already some background in geometry. The course will be given in English.

If you are interested in this seminar, feel free to contact one of us by e-mail. We will hold a zoom meeting (Vorbesprechung) for interested students on Wednesday, February 10 at 12:00, and a second meeting on Wednesday, February 17 at 12:30 where we fix the schedule. Please send us an e-mail if you want to attend this meeting. It is still possible to join the seminar.

Prerequisites. You should be familiar with (or interested in) group theory, general topology, metric spaces, CAT(0) spaces, fundamental groups, simplicial and CW complexes.

Course structure. The course will be run as a seminar course, where the participants will present different topics from the course material. Depending on the number of participants, you will have the chance to present once or multiple times. In order to assist preparation of your presentations and better understanding of the material, we will work in pairs.

Topics. In this course we aim to understand the relationship between group theory and geometry. We will explore the particular case of reflection groups and Coxeter groups and their connections to nonpositively curved spaces. The main reference for the course will be the book “The Geometry and Topology of Coxeter groups” by Michael W. Davis.

Literature
Davis, The Geometry and Topology of Coxteter groups.
Thomas, Geometric and Topological Aspects of Coxeter Groups and Buildings. (You can access the book from the WWU servers, or with vpn)

The course will take place Friday 10:00 - 11:00 via zoom.

Talk 1. (Philip Möller | 23.04.2021) (Chapter 2: Sections 2.1, 2.2 - Group actions, Cayley graphs, word metrics) In geometric group theory we study various topological and metric spaces on which a group G acts. The first of these is the group itself with the discrete topology. The next space of interest is the “Cayley graph.” We will prove that the Cayley graphs for G can be characterized as G-actions on connected graphs which are simply transitively on the vertex set. Similarly, one can define a “Cayley 2-complex” for G to be a simply connected, two dimensional cell complex with a cellular G-action which is simply transitive on its vertex set. These one- and two-dimensional complexes are useful to study the group G.
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Talk 2. (Leon Pernak | 30.04.2021) (Chapter 3: Sections 3.1, 3.2 - Dihedral groups and reflection systems) We define dihedral groups and reflection systems and study their properties.
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Talk 3. (Jonas Morschhausen | 07.05.2021) (Chapter 3, 4: Sections 3.3, 3.4, 3.5, 4.1, 4.2, 4.4 - Coxeter systems and their combinatorial properties) We define Coxeter systems and show that the Cayley graph of a Coxeter system is a reflection system. We also discuss some combinatorial results about Coxeter groups, including special subgroups and subgroups generated by reflections.
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Talk 4. (Linus Kramer | 21.05.2021) (Appendices A, I) Review of cell complexes, polytopes, CAT(κ) spaces, link condition.
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Talk 5. (Julian Blawid | 04.06.2021) (Chapter 5: Sections 5.1, 5.2 - Basic construction U) We discuss the construction of a space U(W,X) associated to a Coxeter systems (W,S), constructed by pasting together copies of X, one for each element of W.
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Talk 6. (Thomas Golüke | 11.06.2021) (Chapter 6: Sections 6.1, 6.2, 6.3, 6.4, 6.5 - Geometric reflection groups) We will study the classical theory of reflection groups on geometries of constant curvature. Let Xⁿ be the n-sphere Sⁿ, the Euclidean n-space Eⁿ, or the hyperbolic n-space Hⁿ. We will discuss the important result that if Pⁿ is a convex polytope in Xⁿ with all dihedral angles integral submultiples of π, then the group W generated by the isometric reflections across the codimension one faces of Pⁿ is a Coxeter group and a discrete subgroup of isometries of Xⁿ, and moreover, Xⁿ = U(W,Pⁿ).
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Talk 7. (Thomas Golüke | 18.06.2021) (Chapter 6: Sections 6.8, 6.9, 6.10 - Simplicial Coxeter groups) We describe a result of Lannér classifying certain Coxeter groups with fundamental chamber a simplex, called simplicial Coxeter groups. This together with earlier results gives a complete classification for Euclidean and spherical cases. Then we discuss Andreev’s theorem to know what happens in the hyperbolic case in dimension 3.
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Talk 8. (Antje Dabeler | 25.06.2021) (Appendix C: The classification of spherical and Euclidean Coxter groups)
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Talk 9. (Jan-Philipp Redlich | 02.07.2021) (Chapter 7: Sections 7.1, 7.2, 7.3, 7.4 - Davis-Moussong complex) Associated to every Coxeter system, there is a natural cell complex Σ with a proper, cocompact W-action. We describe this complex and study geometric and topological properties. For example, Σ is contractible and has a natural CAT(0) metric.
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Talk 10. (Philip Möller | 09.07.2021) (Chapter 12: Sections 12.1, 12.2, 12.3, 12.6 (up to 12.6.6) - Geometry of Σ) We discuss one of the important results that the natural piecewise Euclidean metric on Σ(W,S) is nonpositively curved. The question of whether a polyhedral metric on a cell complex is locally CAT(0) comes down to the question of showing that the link of each vertex is CAT(1). In the right-angled case the fact that the link is CAT(1) is due to Gromov. The corresponding result for the nerve of any Coxeter system was proved by Moussong.
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