Dates: Tuesday, 8:30 - 10:00 (Zeichensaal) and Thursday, 14:15 - 16:00 (SR 10 in Building 2.4)
Lecturers: Prof. Dr. Gabriela Weitze-Schmithüsen
Teaching assistant: Pascal Kattler
The Geometric group theory is a relatively new mathematical discipline which builds interesting connections between group theory and geometry. Its goal is to study groups using geometric methods. There are two main approaches:
- Study how a group acts on a suitable geometric space.
- Consider the group itself as a geometric space.
The interaction of these two fields has lead to a series of mathematical breakthroughs in the last 50 years, among them Gromov's program to classify finitely generated groups, the systematic study of closed three manifolds by William Thurston, the solution of the isomorphism problem of word hyperbolic groups by Sela and the proof of the Haken conjecture by Ian Agol. The theory of automata groups closely links geometric group theory to computer sciences.
In this second part of the lecture we will study central tools of geometric group theory, as for example hyperbolic spaces, invariants for quasi-isometries and the introduction of boundaries for geometric spaces. Furthermore, we put an emphasis on important example of spaces, as for example Teichmüller space T_g with the action of the mapping class group Mod_g on it. Its quotient is the famous moduli space of M_g of closed Riemann surfaces of genus g which plays a prominent role in Geometric group theory but also in Algebraic geometry.
The lecture is a Vertiefungsvorlesung which assumes basic knowledge in algebra and geometry, for example from the lecture Geometric Group Theory I or Algebraic Topology.
- Coarse Geometry: Invariants as Gromov hyperbolicity and the space of ends.
- Hyperbolic Groups
- Central examples from the world of geometric group theory, as for example the mapping class group and Teichmüller space
- Constructions of boundaries
Geometric Group Theory I or Algebraic Topology (recommended).
- Clara Löh: Geometric Group Theory: An Introduction, Springer-Verlag 2017