Growth and generation in SL₂(ℤ∕pℤ)

Abstract

We show that every subset of SL 2 (Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL 2 (Z/pZ), every element of SL 2 (Z/pZ) can be expressed as a product of at most O((log p) c ) elements of A ∪ A −1 , where c and the implied constant are absolute.

Uniform expansion bounds for Cayley graphs of SL2(𝔽p)

“…Recalling (55), we now apply to H the following product theorem in SL 2 (F p ), due to Helfgott [9].…”

“…The novelty of our approach is to derive the upper bound by utilizing the tools of additive combinatorics. In particular, we make crucial use (see §3) of the noncommutative product set estimates, obtained by Tao [26], [27] (Theorems 4 and 5); and of the result of Helfgott [9], asserting that subsets of SL 2 (F p ) grow rapidly under multiplication (Theorem 6). Helfgott's paper, which served as a starting point and an inspiration for our work, builds crucially on sum-product estimates in finite fields due to Bourgain, Glibichuk and Konyagin [3] and Bourgain, Katz, and Tao [4].…”

On the local-global conjecture for integral Apollonian gaskets

“…Bourgain and Gamburd [BG08b] proved uniform spectral gap estimates for Zariski-dense subgroups of SL(2, Z) under the additional assumption that the modulus q is prime. One of the crucial ideas in their paper is the application of Helfgott's triple-product theorem [Hel08]. The result in [BG08b] was generalized in a series of papers [BG08a], [BG09], [BGS10], [Var12], [BV11] and [SGV11].…”

A nilpotent Freiman dimension lemma

“…This is, of course, a necessary step towards exhibiting the sought after nilpotent structure. This idea has intervened several times before under slightly different guises in the classification of approximate groups, be it in Helfgott's original paper [10], in the first two authors' classification of approximate subgroups of compact Lie groups [3], or in our recent paper on the structure of approximate groups in general [5]. It is also closely related to the key idea in the proof of the Margulis lemma in Riemannian geometry, or in the well-known geometric proof by Frobenius and Bieberbach of Jordan's theorem on finite linear groups.…”

“…A number of papers have appeared in the past few years attempting to prove analogues of this result in groups other than the additive group of integers, and in particular in non-commutative groups. See, for example, [1,2,3,4,6,7,9,10,11,12,14,15].…”