Balls in groups: volume, structure and growth

We give sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's theorem on groups with polynomial growth. Precisely, we show that for every non-negative integer d there exists $c=c(d)>0$ such that if $G$ is a group with finite symmetric generating set $S$ containing the identity and $|S^n|\le cn^{d+1}|S|$ for some positive integer $n$ then there exist normal subgroups $H\le\Gamma\le G$ such that $H\subseteq S^n$, such that $\Gamma/H$ is $d$-nilpotent (i.e. has a central series of length $d$ with cyclic factors), and such that $[G:\Gamma]\le g(d)$, where $g(d)$ denotes the maximum order of a finite subgroup of $GL_d(\mathbb{Z})$. The bounds on both the nilpotence and index are sharp; the previous best bounds were $O(d)$ on the nilpotence, and an ineffective function of $d$ on the index. In fact, we obtain this as a small part of a much more detailed fine-scale description of the structure of $G$. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs: percolation theory, random walks, structure of finite groups, scaling limits of finite vertex-transitive graphs.... We obtain some of these applications in the present paper, and treat others in companion papers. Some are due to or joint with other authors.