Seminar: The Geometry and Topology of Coxeter GroupsSommersemester 21Dr. Bakul Sathaye and Prof. Linus Kramer |
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.
Slides 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.
Slides 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.
Slides Talk 4. (Linus Kramer | 21.05.2021) (Appendices A, I) Review of cell complexes, polytopes, CAT(κ) spaces, link condition.
Slides 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.
Slides 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).
Slides 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.
Slides Talk 8. (Antje Dabeler | 25.06.2021) (Appendix C: The classification of spherical and Euclidean Coxter groups)
Slides 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.
Slides 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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