# Groups Acting on Trees Spring 2021

Lecturer
Benjamin Brück
Coordinator
Francesco Fournier-Facio

## Content

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.

### Goal of the course

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

### Brief description

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
• Stallings's theorem on ends of groups

### Prerequisites

Basic knowledge of group theory. The most relevant concepts and results will be recalled during the first exercise class on February 25th.

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.

## Organization

The lecture takes place each Tuesday from 8 to 10, in HG F26.5, starting from February 23rd. Additionally, every second Thursday from 12 to 14, in LFW C4 there will be an exercise class, starting from February 25th.

CHANGE! Due to the current situation, for the moment both the lecture and exercise classes will be held online via zoom: see below.

This class is worth 6 credits. The exam is written, and will take place in the August session.

## Lectures

Due to the current situation, for the moment classes will be held online. The link for accessing the zoom meeting is below, and you will receive by e-mail the password to access the meeting.

Before each lecture, pre-written notes will appear on this page: the lecturer will make reference to those and complete them, so make sure you download them beforehand. After the class, both the final script for the day's lecture and the recording will be available below.

The exercise classes will also be held online via zoom, for the time being. The link for accessing the zoom meeting is below, and you will receive by e-mail the password to access the meeting, which will be available below after the class (the password is the same as for the lectures).

The password to access the recordings is the same as the one for the meetings, followed by the current year with no spaces.

prewritten script recording lecture recording exercise class pdf exercise class
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## Exercises

A new exercise set will appear below every two weeks after the lecture on Tuesday, starting from the first one on February 23. You can hand-in the solution of up to two exercises via SAM-UP until one week and a half later, on Friday. There will be no bonus for handing in solutions, it is just a way for you to check you understand the concepts. There will be no written solutions, but the exercises will be solved during the first exercise class after the hand-in deadline: for instance the first exercise set will be solved during the second exercise class, on March 11.

Together with the exercise sets below you will find a form that will ask you what exercise you did and did not manage to solve. This form is completely anonymous, and it serves as a way to adjust the difficulty of the exercise sets throughout the semester, as well as to identify the hardest exercises that will be talked about more during the exercise class. We kindly ask you to complete the form, even if you did not do the exercises.

exercise set errata due by form
Exercise set 1 Correction to 5 (c) March 5 Form 1
Exercise set 2 - March 19 Form 2
Exercise set 3 $$m, n \in \mathbb{Z} \, \backslash \, \{ 0 \}$$ April 9 Form 3
Exercise set 4 Correction to 2. Also, see above. April 23rd Form 4
Exercise set 5 Correction to 3, 4. Correction to 6. May 7th Form 5
Exercise set 6 The remark at the end was wrong. Correction to 4 May 21st Form 6
Exercise set 7 - June 4th Form 7

## Forum

NEW! After a few requests, we have a opened a subforum on the D-MATH forum website. You are encouraged to discuss exercises and to ask questions.

## 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
• R. C. Lyndon and P. E. Schupp, "Combinatorial group theory". Springer Verlag, 2015. ISBN 3-540-07642-5
• M. J. Dunwoody, "Accessibility and groups of cohomological dimension one". Proceedings of the London Mathematical Society 3.2 (1979): 193-215.
• M. J. Dunwoody, "Cutting up graphs". Combinatorica 2.1 (1982): 15-23.
• B. Krön, "Cutting up graphs revisited – a short proof of Stallings' structure theorem". Groups, Complexity and Topology, vol. 2, no. 2, 2010, pp. 213-221.
• J. Stallings, "Group theory and three-dimensional manifolds". Yale University Press.