Webrelaunch 2020

Geometric Group Theory (Winter Semester 2009/10)

Content

Geometric group theory exhibits interesting relationships between group theory and geometry. Its goal is to study groups with the help of geometric methods, using the two following approaches:

  • Find an appropriate geometric space on which the group acts "nicely".
  • Consider the group itself as a geometric object.

A first example is the Cayley graph of a group with respect to a set of generators. Its vertices are the elements of the group and the edges are given by means of the generators. This construction turns the group into a metric space on which the group acts in a natural way. Another typical example are discrete subgroups of the group \mathrm{SL}_2\mathbb{R} of 2x2-matrices with determinant 1, which act on the upper half plane as isometries for the Poincaré metric.

A prominent example coming from recent research is the Culler-Vogtmann Outerspace \mathrm{CV}_n, which will be one of the topics of the lecture. This is a classification space for finite graphs with an additional structure. The automorphism group of the free group F_n acts on \mathrm{CV}_n. While free groups themselves are relatively simple, their automorphism groups turn out to be not so easy to understand, and there are still open questions to be solved. Culler-Vogtmann outerspace was a useful approach to exhibit some of their properties.

This lecture aims at all, who are interested in algebra and geometry; in particular, students from computer science, who would like to know about relationships between algebraic and graph theoretical and algorithmic methods, and future teachers, who would like to know how to get from simple objects like graphs to deep problems in mathematics, are welcome.

For this lecture, we recommend that you be acquainted with the contents of "Algebra I". Some knowledge in topology is helpful, but not mandatory.

For more information on the lecture and for the problem sheets, see the German version of this page.

Schedule
Lecture: Tuesday 8:00-9:30 AOC 201
Thursday 8:00-9:30 AOC 201
Problem class: Friday 14:00-15:30 Seminarraum S31 Geb. 20.30
Lecturers
Lecturer JProf. Dr. Gabriela Weitze-Schmithüsen
Office hours: no office hours in this semester
Room 1.033 Kollegiengebäude Mathematik (20.30)
Email: weitze-schmithuesen@kit.edu
Problem classes Dr. André Kappes
Office hours: Wann immer ich da bin.
Room 1.036 Kollegiengebäude Mathematik (20.30)
Email: andre.kappes@kit.edu