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 |Sⁿ| cn^d+1|S| for some positive integer n then there exist normal subgroups H G such that H Sⁿ, such that /H is d-nilpotent (i. e. has a central series of length d with cyclic factors), and such that G: g (d), where g (d) denotes the maximum order of a finite subgroup of GLd (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.