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Growth in Permutation Groups and Linear Algebraic Groups Growth in permutation groups and linear algebraic groups H. A. Helfgott Introduction Diameter bounds New work on Growth in permutation groups and linear permutation groups algebraic groups H. A. Helfgott October 2018 Growth in permutation Cayley graphs groups and linear algebraic groups H. A. Helfgott Introduction Definition Diameter bounds G = hSi is a group. The (undirected) Cayley graph New work on Γ( ; ) permutation G S has groups vertex set G and edge set ffg; gag : g 2 G; a 2 Sg. Definition The diameter of Γ(G; S) is −1 diam Γ(G; S) = max min g = s1 ··· sk ; si 2 S [ S : g2G k (Same as graph theoretic diameter.) Growth in permutation How large can the diameter be? groups and linear algebraic groups H. A. Helfgott The diameter can be very small: Introduction Diameter bounds diam Γ(G; G) = 1 New work on permutation groups The diameter also can be very big: ∼ G = hxi = Zn, diam Γ(G; fxg) = bn=2c More generally, G with a large abelian quotient may have Cayley graphs with diameter proportional to jGj. For generic G, however, diameters seem to be much smaller than jGj. Example: for the group G of permutations of the Rubik cube and S the set of moves, jGj = 43252003274489856000, but diam (G; S) = 20 (Davidson, Dethridge, Kociemba and Rokicki, 2010) Growth in permutation The diameter of groups groups and linear algebraic groups Definition H. A. Helfgott diam (G) := max diam Γ(G; S) Introduction S Diameter bounds New work on permutation Conjecture (Babai, in [Babai,Seress 1992]) groups There exists a positive constant c: such that G finite, simple, nonabelian ) diam (G) = O(logc jGj). Conjecture true for PSL(2; p), PSL(3; p) (Helfgott 2008, 2010) PSL(2; q) (Dinai; Varjú); work towards PSLn, PSp2n (Helfgott-Gill 2011) groups of Lie type of bounded rank (Pyber, E. Szabó 2011) and (Breuillard, Green, Tao 2011) But what about permutation groups? Hardest: what about the alternating group An? Growth in permutation Alternating groups, Classification Theorem groups and linear algebraic groups H. A. Helfgott Introduction Reminder: a permutation group is a group of Diameter bounds permutations of n objects. New work on permutation groups Sn = group of all permutations (S = “symmetric”) An = unique subgroup of Sn of index 2 (A = “alternating”) An asymptotic person’s view of the Classification Theorem: The finite simple groups are (a) finite groups of Lie type, (b) An, (c) a finite number of unpleasant things (“sporadic”). Finite numbers of things do not matter asymptotically. We can thus focus on (a) and (b). Growth in permutation Diameter of the alternating group: results groups and linear algebraic groups H. A. Helfgott Introduction Theorem (Helfgott, Seress 2011) Diameter bounds 4 New work on diam (An) ≤ exp(O(log n log log n)): permutation groups Corollary 4 G ≤ Sn transitive ) diam (G) ≤ exp(O(log n log log n)): The corollary follows from the main theorem and (Babai-Seress 1992), which uses the Classification. (As pointed out by Pyber, there is an error in (Babai-Seress 1992), but it has been fixed.) The Helfgott-Seress theorem also uses the Classification. Growth in permutation Product theorems groups and linear algebraic groups H. A. Helfgott Since (Helfgott 2008), diameter results for groups of Lie Introduction type have been proven by product theorems: Diameter bounds Theorem New work on permutation groups There exists a polynomial c(x) such that if G is simple, Lie-type of rank r, G = hAi then A3 = G or jA3j ≥ jAj1+1=c(r): In particular, for bounded r, we have jA3j ≥ jAj1+" for some constant ". Given G = hSi, O(log log jGj) applications of the theorem gives all elements of G. Tripling the length O(log log jGj) times gives diameter 3O(log log jGj) = (log jGj)c. Growth in permutation (Pyber, Spiga) Product theorems are false in groups and linear algebraic groups An. H. A. Helfgott Introduction Example Diameter bounds ∼ G = An, H = Am ≤ G, g = (1; 2;:::; n) (n odd). New work on 3 permutation S = H [ fgg generates G, jS j ≤ 9(m + 1)(m + 2)jSj. groups Related phenomenon: for G of Lie type, rank unbounded, we cannot remove the dependence of the exponent 1=c(r) on the rank r. Powerful techniques, developed for Lie-type groups, are not directly applicable: dimensional estimates (Helfgott 2008, 2010; generalized by Pyber, Szabo, 2011; prefigured in Larsen-Pink, as remarked by Breuillard-Green-Tao, 2011) escape from subvarieties (cf. Eskin-Mozes-Oh, 2005) Growth in permutation Aims groups and linear algebraic groups H. A. Helfgott Introduction Product theorems are useful, and not just because they Diameter bounds imply diameter bounds. They directly imply bounds on New work on spectral gaps, mixing times, etc. permutation groups Our aims are: 1 a simpler, more natural proof of Helfgott-Seress, 2 a weak product theorem for An, 3 a better exponent than 4 in exp((log n)4 log log n), 4 removing the dependence on the Classification Theorem. Here we fulfill aims (1) and (2). L. Pyber is working on (4). Growth in permutation A weak product theorem for An (or Sn) groups and linear algebraic groups Theorem (Helfgott 2018) H. A. Helfgott There are C; c > 0 such that the following holds. Let Introduction −1 A ⊂ Sn be such that A = A and A generates An or Sn. Diameter bounds j j ≥ C(log n)2 New work on Assume A n . Then either permutation groups log jAj log log n C 1+c jAn j ≥ jAj (log n)2 log log n or diam (Γ(hAi; A)) ≤ nC max diam (Γ(G; A0)); A0⊂G G=hA0i where G is a transitive group on m ≤ n elements with no alternating factors of degree > 0:9n. Immediate corollary (via Babai-Seress): Helfgott-Seress bound on the diameter of G = An (or G = Sn), or rather diam G exp(O(log4 n(log log n)2)). Growth in permutation Dimensional estimates and their analogues, I groups and linear algebraic groups The following is an example of a dimensional estimate. H. A. Helfgott Lemma Introduction Let G = SL2(K ), K finite. Let A ⊂ G generate G; assume Diameter bounds −1 A = A . Let V be a one-dimensional subvariety of SL2. New work on 3 1+δ permutation Then either jA j ≥ jAj or groups dim V +O(δ) 1=3+O(δ) jA \ V (K )j ≤ jAj dim SL2 = jAj : A more abstract statement: Lemma Let G be a group. Let R; B ⊂ G, R = R−1. Let k = jBj. Then 1+ 1 2 jRj k [ gRg−1 ≥ : g2B −1 −1 \g2B[feggR Rg If R is special, try to make the denominator trivial. Growth in permutation Dimensional estimates and their analogues, II groups and linear algebraic groups In linear groups, “special” just means “on a subvariety”. H. A. Helfgott What could it mean in a permutation group? Introduction Lemma (Special-set lemma) Diameter bounds Let G be a permutation group. Let R; B ⊂ G, R = R−1, New work on permutation B = B−1, hBi 2-transitive. If R2 has no orbits of length groups > ρn, 0 < ρ < 1, then 2 cρ −1 1+ [g2Br gRg ≥ jRj log n ; 6 where r = O(n ) and cρ > 0 depends only on ρ. This can again be put in the form: for R = A \ special, −1 2 2r+4 either A grows (since ([g2Ar gRg ) ⊂ A ), or R is small compared to A. Idea of proof: produce a small subset D of Br by random walks of length r. Then 2 −1 \g2DgR g is probably trivial (much as in: Babai’s CFSG-free bound on the size of doubly transitive groups). Growth in permutation Building a prefix, I groups and linear algebraic groups H. A. Helfgott Use basic data structures for computations with permutation groups (Sims, 1970) Introduction Given G, write G for the group Diameter bounds (α1,...,αk ) New work on permutation fg 2 G : g(αi ) = αi 81 ≤ i ≤ kg groups (the pointwise stabilizer). Definition A base for G ≤ Sym(Ω) is a sequence of points (α1; : : : ; αk ) such that G(α1,...,αk ) = 1. A base defines a point stabilizer chain G[1] ≥ G[2] ≥ G[3] · · · ≥ [i] with G = G(α1,...,αi−1). Growth in permutation Building a prefix, II groups and linear algebraic groups Choose α1; : : : ; αj greedily so that, at each step, the orbit H. A. Helfgott (A−1A) (α1,...,αi−1) Introduction α i Diameter bounds New work on is maximal. Stop when it is of size < ρn. permutation groups −1 By the special set lemma, (A A)(α1,...,αj ) must be smallish (or else A grows). This implies j (log jAj)=(log n)2. Let Σ = fα1; : : : αj−1g. Because the orbits in all but the last link in the chain are long, the setwise stabilizer 2n (A )Σ, projected to SΣ, is large, and generates A∆ or S∆ for ∆ ⊂ Σ large. We call this the prefix. 2n The pointwise stabilizer (A )(Σ0) restricted to the 0 complement of Σ = Σ [ fαj g is the suffix. 2n The setwise stabilizer (A )Σ0 acts on the suffix by conjugation. Growth in permutation Induction (warning for vegans: Babai-Seress groups and linear algebraic groups uses Classification) H. A. Helfgott The suffix has no orbits of size ≥ ρn. Introduction Diameter bounds What about the group H generated by the setwise 2n New work on stabilizer (A )Σ? If it has no orbits of size ≥ 0:9n, then permutation groups its diameter is not much larger than that of Ab0:9nc, by (Babai-Seress 1992).
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