Groups, trees, and ends. Lesson 4

Что такое группа? Алгебраисты учат, будто это множество с двумя операциями, удовлетворяющими куче легко забываемых аксиом. Это определение вызывает естественный протест: зачем разумному человеку такие пары операций? «Да пропади она пропадом, эта математика» – заключает студент (делающийся в будущем, возможно, министром науки).
В. И. Арнольд. О преподавании математики

This is by far my favourite description of a group! In the following paragraph, Arnold argues that a group exists – in the «ontological» sense – if we can think of it acting on some space. Such space could be the group itself, however the relevant point is that this immediately leads to consider some relevant additional structure, some geometry.
In this series of lectures, we will discover groups through their actions on graphs and in particular on trees, with the help of the very elegant theory developed by Bass and Serre. The leading idea will be that some mild geometrical property can determine a very rigid algebraic structure in the group.
Considering one possible example, to a group one can associate its ends — the «points at infinity» of its graph. For instance, $\mathbb Z$ has two ends (plus and minus infinity), $\mathbb Z^2$ has one (infinity), free group with two generators has infinitely many. A renown theorem by Stallings proposes an algebraic description for groups with infinitely many ends.
A nice proof of this theorem passes through the Bass–Serre theory, describing, how groups can act on trees. We will explain this theory, and, if the time permits, will say a few words on what is left to prove the Stallings theorem, how a group with infinitely many ends can be forced to act on a tree.
Prerequisites: group and graph theory.