In this chapter all the basic notation and concepts are introduced.The notions of nilpotent, solvable, free, linear, finitely generated, and finitely presented groups are defined and examples are provided.Spaces of bounded and Lipschitz harmonic functions are defined, as well as harmonic functions of polynomial growths. Group actions are discussed and convolutions over abstract groups are defined.

The book closes with a brief survey of some of the techniques that have been used to prove the Farrell–Jones conjecture and the Baum–Connes conjecture. We lightly touch on the role of expanders as an obstruction to some of the proof techniques, and the utility of amenability, and its variant, coarse embedding in Hilbert space, as a tool for proving such conjectures.

We prove Breuillard and Green’s theorem that a finite approximate subgroup of a soluble complex linear group G of bounded degree is contained in a union of a few cosets of a nilpotent group of bounded step. We first treat the special case in which G is an upper-triangular group. An important ingredient is Solymosi’s sum-product theorem over the complex numbers, which we state and prove. We introduce some basic representation theory and use it to prove that a soluble complex linear group of bounded degree has a subgroup of bounded index that is conjugate to an upper-triangular group; this is a special case of a result of Mal’cev. We then use this to extend from the upper-triangular case to the general soluble case.

We present Breuillard, Green and Tao’s theorem that a finite approximate subgroup of a complex linear group of bounded degree is contained in a union of a few cosets of a nilpotent subgroup of bounded step. We state two substantial ingredients without proof. The first is a result of Mal’cev and Platinov that a virtually soluble complex linear group of bounded degree has a soluble subgroup of bounded index. The second is Breuillard’s uniform Tits alternative, which states that if a finitely generated complex linear group of bounded degree is not virtually soluble then there exist two free generators of a free subgroup that can be expressed as products of boundedly many generators. The third main ingredient is a result, due independently to Sanders and to Croot and Sisask, that if A is an arbitrary approximate group then there is a relatively large neighbourhood of the identity S with the property that a large power of S is contained in a small power of A; we prove this in full.