Geometric Group Theory (Winter Semester 2007/08)
- Lecturer: JProf. Dr. Gabriela Weitze-Schmithüsen
- Classes: Lecture (1030)
- Weekly hours: 4
- Audience: Mathematics (from 5. semester)
The main goal of geometric group theory is to study properties of groups using geometry by the following two approaches:
- Let the group act on a geometric space and study it by its action.
- Consider the group itself as a geometric object.
A first example is provided by the Cayley graph of a group defined with respect to a system of generators. Its vertices are the elements of the group itself and it makes the group into a metric space. Furthermore, the group acts naturally on it. Unfortunately the graph depends on the chosen generators. An other typical example which we will study are discrete subgroups of SL(2,R) acting on the Poincaré upper half-plane.
More advanced examples are provided by braid groups, the Thompson group, the automorphism group of a free group and mapping class groups. Each of these groups acts naturally on a specific geometric space which itself is an interesting object.
The two approaches described above are closely related by a fundamental theorem of Milnor and Svarc: If a group G acts in a convenient way on a metric space X, then each Cayley graph of G is quasi-isometric to X. Roughly speaking this means that from far away they look the same. We will study in the course properties of groups and more generally of metric spaces that only depend on the quasi-isometry class. One such property is Gromov hyperbolicity, a concept which generalizes classical hyperbolic geometry as well as the notion of trees and R-trees. One can deduce nice algebraic properties from the fact that a group is Gromov hyperbolic.
The knowledge of the course ``Algebra I'' is presumed. Basic knowledge in topology and hyperbolic geometry is helpful but not required.
Some web pages on geometric group theory
Here are some general references for the topic:
- M. Bridson, A. Haefliger: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften. 319. Springer 1999.
- M. Coornaert, T. Delzant, A. Papadopoulos:Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov.(Geometry and group theory. The hyperbolic groups of Gromov). Lecture Notes in Mathematics, 1441. Springer 1990.
- E. Ghys (ed.); A. Haefliger (ed.); A. Verjovsky: Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Reprint of the 1991 ed.
- E. Ghys, P. de la Harpe (ed.): Sur les Groupes Hyperboliques d'après-Mikhael Gromov. (On the hyperbolic groups a la M. Gromov. (French) Progress in Mathematics, 83. Birkh\"auser 1990.
- C. Leininger/A. Reed: Notes on Geometric group theory. Informal notes: http://www.math.uiuc.edu/~clein/.