This chapter provides a full elementary proof of Gromov’s theorem, which states that a finitely generated group has polynomial growth if and only if it is virtually nilpotent. The proof proceeds along the ideas laid forth by Ozawa, using the existence of a harmonic cocycle. Gromov’s theorem is then used to classify all recurrent groups. Also, consequences of harmonic cocycle to diffusivitiy of the walk are shown.

The focus of this chapter is the geometrically defined class of Polish groups of bounded geometry, that form an almost perfect geometric generalisation of the locally compact second countable groups. We provide some canonical examples of such groups and establish a number of basic results that mimic results known from the locally compact case. Most importantly, we provide the appropriate generalisation of Gromov's Theorem equating quasi-isometry of groups with topological couplings.

We present various applications of Breuillard, Green and Tao’s rough classification of finite approximate groups to groups of polynomial growth. We define polynomial, exponential and intermediate growth, and show that these concepts are stable under changes of generating set and passing to subgroups of finite index. We prove Breuillard, Green and Tao’s result that if a ball of large enough radius in a Cayley graph is of size polynomial in the radius then the underlying group is virtually nilpotent. We deduce that all larger balls also have polynomial bounds on their sizes. We guide the reader in the exercises to Breuillard and Tointon’s results that a finite group of large diameter admits large virtually nilpotent and virtually abelian quotients. We also prove the same authors’ result that a finite simple group has diameter bounded by a small power of the size of the group. We prove an isoperimetric inequality for finite groups due to Breuillard, Green and Tao. Finally, we give a brief high-level introduction to applications of approximate groups to the construction of expanders.