## Benjamin Brück: Katalogdaten im Frühjahrssemester 2021 |

Name | Herr Dr. Benjamin Brück |

Departement | Mathematik |

Beziehung | Dozent |

Nummer | Titel | ECTS | Umfang | Dozierende | |
---|---|---|---|---|---|

401-4206-17L | Groups Acting on Trees | 6 KP | 3G | B. Brück | |

Kurzbeschreibung | 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. | ||||

Lernziel | Learn basics of Bass-Serre theory; get to know concepts from geometric group theory. | ||||

Inhalt | 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 | ||||

Literatur | 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 | ||||

Voraussetzungen / Besonderes | Basic knowledge of group theory; being familiar with fundamental groups (e.g. the Seifert-van-Kampen Theorem) and covering theory is definitely helpful, although not strictly necessary. In particular, the standard material of the first two years of the Mathematics Bachelor is sufficient. |