Module 24-M-GT-GGT Introduction to Geometric Group Theory

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master the basic contents and methods of Geometric Group Theory, in particular they can independently carry out complex proofs in this area requiring a high level of mathematical expertise. Students are able to define central concepts of the theory (e.g. Cayleygraph, quasi-isometry, hyperbolic group, growth, properly discontinuous action) and apply them correctly. They know leading examples of the theory (e.g. free groups,
matrix groups over the integers, braid groups, reflection groups) and can use these examples to illustrate concepts and theorems. E.g. students can determine the growth functions of free groups.

Furthermore, the students recognise further-reaching connections to mathematical facts already acquired. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study. In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Geometric Group Theory.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following content of teaching from the field of Geometric Group Theory is compulsory:

• groups as metric spaces (word metric, Cayley graph, hyperbolicity)
• properly discontinuous actions and the Schwarz-Milnor theorem
• quasi-isometries and QI invariants / QI non-invariants
• CAT(0) geometry and CAT(0) groups
• boundaries of groups
• important examples: free groups, free abelian groups, matrix groups over arithmetic rings, groups of (outer) automorphisms, braid groups, reflection groups, the groups of Thompson and Grigorchuk

In addition, the following contents of teaching can be covered, for example:

• ameanability and property (T)
• finiteness properties
• graphs of groups

This module prepares the content of a master's thesis.

Algebra and Geometry and Topology

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