Combinatorial Group Theory
Code
11638
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Teaching language
Português
Objectives
By the end of this curricular unit the student should have acquired knowledge, skills and competences that allow him to:
1) profound knowledge of the fundamental concepts of combinatorial group theory, such as free group, presentation, free product and graph product, Van Kampen diagram, HNN extensions and free amalgamated product, and hyperbolic group;
2) being able to enounce and sketch the proof the basic results of the theory, such as the Seifert-Van Kampen Theorem, the Nielse-Schreier theorem, the Kurosh theorem or the Svarc-Milnor Lemma;
3) know how to apply the fundamental results of the theory;
4) getting acquaintance with the recent trends of combinatorial and geometric group theory.
Subject matter
1. Free groups, their properties and their subgroups via Stallings subgroup graphs. Representations of free groups.
2. Groups given by generators and relations. The calculus of presentations and the method of Reidemeister and Schreier. Cayley graphs and the word metric. Tietze transformations. Van Kampen diagrams and Van Kampen Theorem.
3. Hyperbolic groups, quasi-isometries and quasiconvex subgroups.
4. Groups actions on sets. Groups actions on graphs by isometries and Bass-Serre theory, amalgamated free products and HNN extensions, graphs of groups and group actions on simplicial trees.
5. One or more of the following topics: groups with a single defining relator; the study of isoperimetric inequalities; the Novikov-Boone Theorem; the Higman Embedding Theorem; Grigorchuk groups of intermediate growth; automatic groups, etc.
Bibliography
G. Baumslag, Topics in combinatorial group theory, Springer, 1993.
O. Bogopolski , Introduction to group theory, EMS, 2008.
W. Dicks and M. Dunwoody, Groups acting on graphs, Cambridge University Press, 1989.
R. Lyndon and P. Schupp, Combinatorial group theory, Springer, 2001.
P. de la Harpe, Topics in geometric group theory. Chicago lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.
J. Rotman, An introduction to the theory of groups, Springer, 4th Ed, 1995.