Growth in finite groups and the Graph Isomorphism Problem Dissertation for the award of the degree \Doctor rerum naturalium" (Dr. rer. nat.) of the Georg-August-Universit¨atG¨ottingen within the doctoral program \Mathematical Sciences" of the Georg-August University School of Science (GAUSS) submitted by Daniele Dona from Torino G¨ottingen,2020 Thesis committee Harald Andr´esHelfgott Mathematisches Institut Georg-August-Universit¨atG¨ottingen Valentin Blomer Mathematisches Institut Universit¨atBonn Members of the Examination Board First reviewer: Harald Andr´esHelfgott Mathematisches Institut Georg-August-Universit¨atG¨ottingen Second reviewer: Laurent Bartholdi Mathematisches Institut Georg-August-Universit¨atG¨ottingen Further members of the Examination Board Valentin Blomer Mathematisches Institut Universit¨atBonn Axel Munk Institut f¨urMathematische Stochastik Georg-August-Universit¨atG¨ottingen Matthew Tointon Department of Pure Mathematics and Mathematical Statistics University of Cambridge P´eterVarj´u Department of Pure Mathematics and Mathematical Statistics University of Cambridge Date of the oral examination: 17 July 2020. Contents Preface 5 Acknowledgements 7 1 General introduction 11 1.1 Growth in groups, Cayley graphs, diameters . 11 1.2 Finite simple groups . 15 1.3 Babai's conjecture . 19 1.4 Other results on growth and diameter . 22 1.5 The graph isomorphism problem . 26 1.6 Babai's algorithm . 29 2 The Weisfeiler-Leman algorithm and the diameter of Schreier graphs 33 2.1 Introduction . 33 2.2 The upper bound . 38 2.3 The lower bound . 42 2.4 The case of Cayley graphs . 45 2.5 Concluding remarks . 48 3 Short expressions for cosets of permutation subgroups 51 3.1 Standard definitions . 52 3.2 Main theorem: statement . 53 3.3 Elementary routines . 56 3.4 Major routines . 61 3.5 The algorithm . 64 3.5.1 The algorithm, assuming CFSG . 71 3.5.2 The algorithm, not assuming CFSG . 79 3.6 Main theorem: proof . 82 3.7 Concluding remarks . 88 4 Slowly growing sets in Aff(Fq) 91 4.1 Introduction . 92 2 4.2 Number of directions in Fq ...................... 94 4.3 Growth in Aff(Fq)........................... 99 3 4.4 Concluding remarks . 101 5 Diameter bounds for products of finite simple groups 103 5.1 Main theorem . 103 5.2 Preliminaries . 105 5.3 Proof of the main theorem . 106 5.4 Concluding remarks . 109 6 Towards a CFSG-free diameter bound for Alt(n) 111 6.1 Background and strategy . 112 6.2 Tools . 114 6.3 Main theorem . 117 6.4 Concluding remarks . 127 Bibliography 129 4 Preface The present thesis embraces two major areas of mathematics, namely group the- ory (especially growth in finite groups) and graph theory (especially the graph isomorphism problem). Chapter 1 serves as a somewhat lengthy introduction to both, with x1.1-1.2- 1.3-1.4 focusing on growth in groups and x1.5-1.6 on graph isomorphisms. The next two chapters are mostly graph-theoretic, although they bear many connections to growth in groups as well. Chapter 2 is based on the author's published article [Don19c]; its main results are Theorem 2.1.6, Theorem 2.1.7 and Theorem 2.1.8. Chapter 3 is based on the author's preprint [Don18]; its main result is Theorem 3.2.1. The two chapters that follow are entirely on the topic of growth in groups. Chapter 4 is based on the author's preprint [Don19b]; its main results are Theo- rem 4.1.3 and Theorem 4.1.4. Chapter 5 is based on the author's preprint [Don19a]; its main result is Theorem 5.1.1. Finally the last one, Chapter 6, is firmly rooted into both areas at once: more precisely, graph-theoretic tools intervene in group-theoretic problems; its main result is Theorem 6.3.6, dependent on Conjecture 6.3.4. Notation. Any and all notations hold unless otherwise stated. We adopt the big O notation for describing orders of magnitude. If f; g are some real-valued functions, we say f(x) = O(g(x)) to mean that there exists a constant C > 0 such that jf(x)j ≤ Cg(x) for all x in the intersection of the domains of f; g; since we are almost always considering f to have domain N and codomain inside R≥0, in those cases it suffices to say that f(x) ≤ Cg(x) for all x large enough. We also use f(x) = o(g(x)) to mean that for all C > 0 and all x large enough (depending on C) we have f(x) < Cg(x). Finally, f(x) = Ω(g(x)) means that there exists C > 0 such that f(x) ≥ Cg(x) for all x large enough: in this, we follow Knuth's definition of the symbol instead of Hardy and Littlewood's convention (see Knuth's own letter to the editor [Knu76], where \O" is incidentally revealed to be an omicron!). If we want to emphasize that the constant C in the notations above depends on other parameters (say n; k), we write them as indices to the symbol (say On;k(g(x))). Many other authors, especially of the number theory school, use also Vinogradov's and notation: the author appreciates the fact that essentially the same symbol facing two directions can do the job of both O(·) and Ω(·), but he also needs to write things like eO(x), for which Vinogradov provides no solution; thus, no will be used. 5 For the set f1; 2; : : : ; ng of natural numbers from 1 to n, we often write [n] for brevity, as is common in the literature regarding permutation groups; to be clear, the author subscribes to the convention that 0 2 N, but 0 has usually little space in the context of permutations. For us, p denotes a prime number, and q denotes a prime power. If X is a finite set, the set of permutations of X is denoted by Sym(X), and the set of even permutations by Alt(X); in particular, we write Sym(n); Alt(n) for Sym([n]); Alt([n]) (we will not use the notations Sn;An and Symn; Altn that frequently occur elsewhere). As for algebraic groups, say the special linear group, we use the notation SLn(Fq) instead of the equally widespread SL(n; Fq) and SL(n; q). a We use loga x to denote the logarithm of x in base a, and log x to denote (log x)a. Since we will not be using longer expressions than log log x, there is no need to use either notation for the iterated logarithm, as some authors do (and for good reasons need to do). About identity elements, notation varies with the context. For general groups, like in x1 and x5, we use e to denote the group's identity; for permutation groups of degree n like in x3 we use Idn, while for the matrix groups in x4 we use Id without index since we work only with 2×2 matrices. In x2, where several identities coexist, we try and use distinct notations: e for general groups, Idn for n × n matrices, IdX for automorphisms on the object X. In x6, where we work with permutation groups but their identity elements are encountered only in their quality of group identities, we use e. Finally, since we abundantly use several terms describing orders of magnitude, which may be unfamiliar to the readers, we collect them here: k • f(x) is quasipolynomial in x when f(x) ≤ eC log x for some absolute con- stants C; k > 0; • f(x) is polylogarithmic in x when f(x) ≤ C logk x for some absolute constants C; k > 0. 6 Acknowledgements One can always aspire to be like the Laplacian intelligence, embracing the whole universe in her thought, for whom nothing would be uncertain, the future as well as the past always present in her eyes1. And one can always delude oneself about having an arm strong enough to pull Leviathan out of the water2; and being able to produce a book on which everything is contained as a whole, and from it judge the world3, as if Protagoras had actually meant a very specific man to be the measure of all things4. These solitary God-worthy feats one can surely believe to be within reach, and fancy oneself the fixed terminus of eternal counsel5, the skies and the stars revolving and revolving under the push of this one finger6; and expect that all that is in the world bends slowly and surely to one's will, weighed by the doom of one's thought7. But mit der Dummheit k¨ampfenG¨otterselbst vergebens8. And even as I pile quotations in the hope of building a stand high enough for my sense of self- importance, and lose myself in my own mighty station and my stupendous brain9 until disillusion comes too late to evade my own ruinous shipwreck10, I must recognize the many other people around me that have contributed to making this achievement of mine possible, people without whom I would not be standing where I am, and to whom I owe in different ways all that I have today. Half-joking ram- blings aside, I often hesitate, get discouraged, and doubt that the position that I am in is well-deserved11; that is why people who supported me throughout this whole endeavour, and continue to do so even now, deserve all the gratitude that I can muster and should get their due credit. 1Pierre-Simon Laplace, Essai philosophique sur les probabilit´es, p. 4 (1840 edition). 2Anonymous, -Iyy¯ob, 41:1.