Geometric Group Theory

 

Geometric Group Theory

 

Dates:

• Lecture: Wednesday, 10 am (c.t.) and Thursday, 2pm (c.t.) in SR 10 (Building E 2.4)
• Exercise Section: to be anounced

The Team:

• Lecturer: Prof. Dr. Gabriela Weitze-Schmithüsen (weitze [at] math.uni-sb.de)
• Assistant: Pascal Kattler (nachname [at] math.uni-sb.de)
• Tutor: Robin Lahni

Prerequisites:

Linear Algebra I und II and Algebra (recommended).
 

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Topic:

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.

This lecture is a first introduction to geometric group theory. We will introduce Cayley graphs, the theory of quasi-isometries and the so-called coase geometry. Some highlights will be the theorem of Schwarz and Milnor which relates geometric properties of the group to  geometric spaces on which it acts in a good way, as well as examples of groups which play an eminent role in current research. This will offer insights into a rich world between algebra and geometry.

The lecture is a new Stammvorlesung which gives access to the mathematical disciplines of the two lecturer. It is aimed at bachelor and master students. With its algorithmic aspects it can also be an interesting lecture for students in computer sciences.

 

Contents:

• Free groups, presentations of groups, Cayley graphs
• Fundamental groups and covering theory
• Coarse geometry, quasi-isometries ad the theorem of Milnor and Schwarz
• Gromov hyperbolicity
• Optional: Growth of groups, space of ends, Fuchsian groups, Example of geometric spaces as Teichmüller space, translation surfaces and outer space.

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Literature:

• Pierre de la Harpe: Topics in Geometric Group Theory, Chicago Lectures in Mathematics 2003

• Clara Löh: Geometric Group Theory, Springer-Verlag 2017

Supplementary Literature:

• Otto Forster: Lectures on Riemann Surfaces (e.g. for theory of coverings, universal covering)
• Roger C. Lyndon , Paul E. Schupp: Combinatorial Group Theory (e.g. for group presentations, free products with amalgamation, HNN extensions)
• John Stillwell: Classical Topology and Combinatorial Group Theory (e.g. for fundamental groups, Theorem Seifert/van Campen, classification of closed surfaces, Dehn algorithm for surface groups)
• L. Christine Kinsey: Topology of Surfaces (e.g. for classification of closed surfaces)