This chapter is devoted to the study of the space of bounded harmonic functions and the Liouville property. We start with the entropic criterion for the Liouville property. We then investigate the relationship of the Liouville property with amenability, speed of the random walk, and coupling of exit measures.The central example of lamplighter groups is studied.

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.

This chapter is devoted to proving the Milnor–Wolf theorem, which states that a finitely generated solvable group has polynomial growth if and only if it is actually virtually nilpotent.

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.

Here, we dive deeper into the realm of reversible Markov chains, via the perspective of network theory. The notions of conductance and resistance are defined, as well as voltage and current, and the corresponding mathematical theory.The Laplacian and Green function are defined and their relation to harmonic functions explained. The chapter culminates with a proof (using network theory) that recurrence and transience are essentially group properties: these properties remain invariant when changing between different reasonable random walks on the same group (specifically, symmetric and adapted with finite second moment).

This chapter dives into the theory of (discrete time) martingales.The optional stopping theorem and the martingale convergence theorem are proved.These are used to provide some initial results regarding random walks on groups and bounded harmonic functions. Specifically, the random walk on the integer line is shown to be recurrent. Also, it is shown that the space of bounded harmonic functions is either just the constant functions or has infinite dimension.

The Choquet–Deny theorem states that any random walk on a nilpotent group is Liouville. This theorem is presented and proved. We then present a recent result from 2018 by Frisch, Hartman, Tamuz, and Vahidi-Ferdowski, that these are basically the only such examples.

In this chapter we start applying the tools developed in Part I to study random walks.The notion of amenable groups is defined, and Kesten’s criterion for amenable groups is proved. We then move to define the notion of isopermitric dimension. Inequalities relating the volume growth of a group to the isoperimetric dimension and to the decay of the heat kernel are proved.