The first part of this course is a basic introduction to the geometry and combinatorics of countable infinite groups, and it follows a selection of the chapters in the book [4]. We will start by recalling the notion and basic properties of Cayley and Schreier graphs, and analysing some fundamental examples: free abelian groups, free groups, Baumslag-Solitar groups, lamplighter groups, and surface groups. Using these examples we will introduce some fundamental notions, such as amenability, non-amenability, ends of a group, and the word problem.

In the second part we will start by recalling the ”Banach-Tarski paradox”, and how it leads to various ”equidecomposability problems” which have been studied over the last century. In such problems we are given two sets A and B (e.g. a ball and a cube of the same volume in R 3 ), and we are asking whether it is possible to partition A into sufficiently regular sets and move those pieces to assemble B (see [5] for more details). We will see how the group- theoretic Property (T) and the ergodic-theoretic equidistribution results allow to prove various equidecomposability results.

Both Property (T) and equidistribution are fundamental notions whose applications go far beyond equidecomposability problems, and we will discuss some of them (for example the construction of expander graphs). A standard reference for property (T) is the book [1], and the equidistribution result which we will discuss is the famous theorem of Laczkovich [3].

A very convenient context for relating equidecomposability problems and e.g. Property (T) is provided by measured equivalence relations and graphings. As such we will spend some time to discuss those notions, and go through relevant examples and basic properties (such as treeability and cost, see [2]). We will also discuss some interesting open problems about infinite groups and graphings, e.g. Gromov’s question about sofic groups, Aldous-Lyons conjecture, and some of Gaboriau’s questions about cost.

My intention is to make this course interesting and accessible to a wide variety of audience. I will assume only standard undergraduate courses as prerequisites. The course will run in english, once per week for 90min in the Summer semester.