Submission date: 4 March 2024

Abstract:

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

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Full text arXiv 2403.02485: pdf, ps.