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Growth in Linear Groups 3 GROWTH IN LINEAR GROUPS SEAN EBERHARD, BRENDAN MURPHY, LASZL´ O´ PYBER, AND ENDRE SZABO´ Abstract. We prove a conjecture of Helfgott and Lindenstrauss on the struc- ture of sets of bounded tripling in bounded rank, which states the follow- ing. Let A be a finite symmetric subset of GLn(F) for any field F such that |A3|≤ K|A|. Then there are subgroups H E Γ ≤ hAi such that A is covered by KOn(1) cosets of Γ, Γ/H is nilpotent of step at most n − 1, and H is contained in AOn(1). Contents 1. Introduction 1 2. Toolbox 4 3. Virtually soluble linear groups 7 4. Reduction to soluble-by-Lie* 10 5. The soluble radical of the perfect core 12 6. Trigonalization and removing small roots 15 7. Pivoting 17 8. Growth of bilinear images 18 9. The no-small-roots case 20 References 22 1. Introduction In this paper we aim to characterize sets of bounded tripling in GLn(F), where n is bounded and F is an arbitrary field. To be precise, a finite set A ⊆ GLn(F) is said arXiv:2107.06674v1 [math.GR] 14 Jul 2021 to be K-tripling if |A3|≤ K|A|. This notion is largely the same as that of a finite K-approximate group, which is a symmetric set A such that A2 is covered by at most K translates of A. Prototypical examples include subgroups and progressions {gn : |n| ≤ N}, as well as certain nilpotent generalizations of progressions called nilprogressions, such as the following standard example. SE has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 803711). BM was funded by The Leverhulme Trust through Leverhulme grant RPG 2017-371. LP was supported by the National Research, Development and Innovation Office (NKFIH) Grant K115799, ESz was supported by the NKFIH Grants K115799 and K120697. The project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 741420). 1 2 SEAN EBERHARD, BRENDAN MURPHY, LASZL´ O´ PYBER, AND ENDRE SZABO´ Example 1.1 (the Heisenberg nilprogression). Let 1 x z 2 A = 0 1 y : |x|, |y|≤ N, |z|≤ N ⊆ GL3(R). 0 0 1 Then |A3|≤ 100|A|. Essentially, the most general approximate group is an extension of a subgroup by a nilprogression: this is the content of the following general structure theorem for approximate groups proved by Breuillard, Green, and Tao [BGT2]. Theorem 1.2 (Breuillard, Green, Tao [BGT2]). Let A be a K-approximate sub- group of some group G. Then there are subgroups H E Γ ≤ G with the following properties: (1) A is covered by OK (1) cosets of Γ, (2) Γ/H is nilpotent of rank and step at most OK (1), (3) H is contained in A4. The method of proof is based on nonstandard analysis as well as the solution to Hilbert’s fifth problem, and there seems to be no way to deduce an explicit bound for the number of cosets of Γ required to cover A (the bounds on the rank and step of Γ/H are explicit and benign: see [BGT2, Remark 1.9]). Moreover, in [BT,E] the following example is given which shows that this bound must be at least Kc log log K in general, i.e., not polynomial. n n Example 1.3. Let G = Z ⋊ Sn and let A = [−r, r] Sn for any r>n!. It can be shown that A is a 2n-approximate group and not covered by fewer than n! cosets of any finite-by-nilpotent subgroup. Moreover A is not covered by fewer than n!/24n/3 cosets of any finite-by-soluble subgroup. However in bounded rank things are different. In this paper we prove a version of the above theorem in bounded rank with polynomial bounds. A result of this form was originally predicted by Helfgott [H1,T] and Lindenstrauss [personal com- munication]; it was later formulated precisely as a conjecture in [GH] as well as [H2]. Theorem 1.4. Let F be an arbitrary field. Let A ⊆ GLn(F) be a finite symmetric subset such that |A3|≤ K|A|. Then there are subgroups H E Γ ≤ hAi such that (1) A is covered by KOn(1) cosets of Γ, (2) Γ/H is nilpotent of step at most n − 1, (3) H is contained in AOn(1). Note that without loss of generality H can be taken to be γn(Γ), where (γm(G))m≥1 is the lower central series of Γ. In characteristic zero, Theorem 1.4 was first established by Breuillard, Green, and Tao [BGT1]. The case of finite prime fields was established by Gill and Helfgott (jointly with the third and fourth authors) in [GH]. In this paper we consider all fields uniformly, mostly by reducing to the finite case. In the prime finite field case, Gill and Helfgott additionally proved that Γ can be taken to be normal in hAi. We do not know whether this is true in general, though it is true if we only demand that Γ/H be soluble. This result is established in the course of the proof of Theorem 1.4, and was previously announced in [PS2]. GROWTH IN LINEAR GROUPS 3 Theorem 1.5. In the situation of Theorem 1.4 there are subgroups P E Γ, both normal in hAi, such that (1) A is covered by KOn(1) cosets of Γ, (2) Γ/P is soluble of derived length O(log n), (3) P is perfect and contained in a translate of A3. This result extends (and depends on) the Product Theorem for finite simple groups, obtained independently by Breuillard, Green, Tao [BGT1] and Pyber, Szab´o[PS3]. Theorem 1.6 ([PS3, Theorem 2], see also [BGT1, Corollary 2.4]). Let L be a finite simple group of Lie type of rank r and A a generating set of L. Then either (1) |A3| > |A|1+ε, where ε = ε(r) depends only on r, or (2) A3 = L. 1.1. Roadmap. The structure of the paper is as follows. Section 2 recalls some tools that may be familiar to experts: basic group the- ory and arithmetic combinatorics (or nonabelian additive combinatorics), quasi- randomness, and the action of p′-groups on p-groups. Section 3 contains structural results for virtually soluble linear groups. A result of Mal’cev states that soluble linear groups of rank n have a trigonal subgroup of index On(1); hence virtually soluble linear groups of rank n are virtually trigonal- izable, but there is no bound on the index of the trigonal subgroup. Using results of Platonov and Weisfeiler (or Larsen and Pink), we show that a virtually soluble linear group of rank n and characteristic p contains a subgroup of index On(1) that is soluble-by-Lie∗(p) (that is, an extension of a direct product of finite simple groups of Lie type in characteristic p by a soluble group). A few other facts about soluble-by-Lie∗(p) linear groups are also proved. Section 4 contains a result, Theorem 4.1, from a preprint of the last two authors, which is a preliminary version of Theorem 1.5. This result states that if A ⊆ GLn(F) has small tripling, then A is covered by polynomially many cosets of a soluble-by- Lie∗(p) group Γ (where p = char F), A6 covers the Lie∗(p) quotient Γ/ Sol(Γ) in the sense that hAi ∩ Γ projects surjectively onto Γ/ Sol(Γ), and this Lie∗(p) quotient is quasi-random. Section 5 contains the proof of Theorem 1.5. Given a set of small tripling A, the results of the previous section produce a subgroup Γ of hAi so that Γ modulo a soluble normal subgroup is covered by a small power of A. We wish to produce a (perfect) normal subgroup P E Γ so that P is covered by a small power of A, and Γ/P is soluble. In a sense, we must move the perfect group from on top of the soluble group to below it. Our candidate for P is the perfect core of Γ; that is, P is the last term of the derived series of Γ. The quotient Γ/P is clearly soluble, so it remains to generate P . We show that P/ Sol(P ) is quasi-random, and that Sol(P ) has a nilpotent subgroup N of index On(1). This reduces the proof of Theorem 1.5 to that of Proposition 5.3, which uses the quasi-randomness of P/ Sol(P ) and its action on N to quickly generate P . Theorem 1.5 reduces the study of subsets A of GLn(F) of small tripling to those where hAi is soluble. Thus to prove Theorem 1.4, it remains to study sets of small tripling that generate soluble groups. 4 SEAN EBERHARD, BRENDAN MURPHY, LASZL´ O´ PYBER, AND ENDRE SZABO´ Section 6 reduces the problem to studying trigonal groups, so without loss of generality, we may assume that A is contained in the (Borel) subgroup of upper triangular matrices in GLn(F). The next two sections contain “growth” results, which are the main tools we need to prove the trigonal case. Section 7 contains a result of Gill and Helfgott concerning the action of an abelian group on another group; this can be seen as an extremely general version of a “sum-product theorem”.
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