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Combinatorial Group Theory

Objectives

By the end of this curricular unit, the student should have acquired knowledge, skills, and competencies 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 of the basic results of the theory, such as the Seifert-Van Kampen Theorem, the Nielsen-Schreier theorem, the Kurosh theorem, or the Svarc-Milnor Lemma;

3) know how to apply the fundamental results of the theory;

4) getting acquainted with the recent trends of combinatorial and geometric group theory.

General characterization

Code

12986

Credits

9.0

Responsible teacher

António José Mesquita da Cunha Machado Malheiro

Hours

Weekly - 4

Total - 56

Teaching language

Inglês

Prerequisites

General knowledge about algebra and more specific knowledge about group theory.

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.

Teaching method

The adopted teaching strategy is based on lectures in which it is intended to expose fundamental concepts and track students'' work.

Through scheduled contact hours we intend to make an oral presentation of the main topics, followed by small representative examples that allow a better understanding of theoretical concepts.

 

The students have exercises that should be solved individually, and during the hours of support/doubts discuss solutions with the teacher.

 

Evaluation method

The student will have to solve every two weeks exercises proposed by the teacher.

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.