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Groups Acting on Trees
Last Updated: 2026-02-05 15:54:12
Abstract
As a main theme, we will see how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure.
Objective
Learn basics of Bass-Serre theory; get to know concepts from geometric group theory.
Content
As a mathematical object, a tree is a graph without any loops. It turns out that if a group acts on such an object, the algebraic structure of the group has a nice description in terms of the combinatorics of the graph. In particular, groups acting on trees can be decomposed in a certain way into simpler pieces.These decompositions can be described combinatorially, but are closely related to concepts from topology such as fundamental groups and covering spaces. This interplay between (elementary) concepts of algebra, combinatorics and geometry/topology is typical for geometric group theory. The course can also serve as an introduction to basic concepts of this field. Topics that will be covered in the lecture include: - Trees and their automorphisms - Different characterisations of free groups - Amalgamated products and HNN extensions - Graphs of groups - Kurosh's theorem on subgroups of free (amalgamated) products
Resources
Literature
J.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9 O. Bogopolski. Introduction to group theory. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. x+177 pp. ISBN: 978-3-03719-041-8 C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Learning Materials (Links)
- Main link
- Homepage of the lecture
General Information
- Language
- English
- Levels
- BSC , DR , MSC
Examination
- Type
- session examination
- Mode
- written 120 minutes
- Aids
- None.
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture with exercise |
Groups Acting on Trees
Groups are selected in myStudies.
lectures Tue 8-10
exercises Thu 12-14 two-weekly
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3 h weekly |
Offered In
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Electives (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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Doctoral Department of Mathematics (More Information at: The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM. WARNING: Do not mistake ECTS credits for credit points for doctoral studies!)
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Graduate School (Official website of the Zurich Graduate School in Mathematics:)
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