Thin Groups and the Affine Sieve

Transcription

1 Thin Groups and the Affine Sieve Peter Sarnak Mahler Lectures 2011

2 SL n (Z) the group of integer n n matrices of determinant equal to 1. It is a complicated big group It is central in automorphic forms, number theory, geometry... It satisfies some basic properties when reduced modulo q: (1) Strong Approximation (Chinese remainder theorem) SL n (Z) πq SL n (Z/qZ) is onto. There is a quantification of this that is also fundamental.

3 Fix a finite generating set S of SL n (Z) (assumethatitis symmetric, s S s 1 S). Form the finite congruence graphs X q =(SL n (Z/qZ), S), vertices are elements of SL n (Z/qZ), edges g sg, s S. X q is connected (by strong approximation), X q is S -regular.

4 (2) Super-strong Approximation The X q s are an expander family, i.e. if the eigenvalues of the adjacency matrix satisfy with ε 0 > 0 (independent of q!) spectral gap. S = λ 1 > λ 2 λ 3... λ N λ 2 S ε 0 = the graphs X q are very highly connected, random walk on X q with generators S is rapidly mixing,....

5 (2) follows from automorphic forms. If Γ(q) =ker(a A (mod q)), consider L 2 (Γ(q) \ SL n (R)), and in particular the Ramanujan Selberg Conjectures about which a lot is known. (If n 3, one can also use property T ). More generally, if G is a semisimple simply-connected group defined over Q, then both (1) and (2) continue to hold for Γ = G(Z) (assumeg(r) has no compact factors). (2) due to Burger Sarnak Clozel property tau.

6 For many applications, one needs these fundamental properties for general Γ SL n (Z). let G = Zcl(Γ), the Zariski closure of Γ. The smallest algebraic matrix group to contain Γ. Its equations are over Q. So G is a familiar and a well understood object. Definition If Γ is of infinite index in G(Z), we say Γ is thin. In general, the question of whether Γ is thin has no decision procedure (Mikhailova 1958). Ubiquity of Thin Groups: (A) Fix 2 and choose A 1,...,A at random in SL n (Z) by taking them from a big ball A j X, j =1,...,. Thenwith probability tending to 1 as X, Γ = A 1,...,A is Zariski dense in SL n, it is thin and free (in fact Schottky ). R. Aoun (2010).

7 (B) Diophantine geometric constructions typically yield thin groups. E.g.: Integral Apollonian packings: F (x 1, x 2, x 3, x 4 ) = 2(x x x x 2 4 ) (x 1 + x 2 + x 3 + x 4 ) 2 G = O F, the orthogonal group of F O F (Z) GL 4 (Z) A = Apollonian group, A = S 1, S 2, S 3, S S 1 = , S 2 = , S 3 = , S 4 = A O F (Z), A is thin!

8 Peter Sarnak Mahler Lectures 2011 Thin Groups and the Affine Sieve

9 If a =( 11, 21, 24, 28), then the orbit O a = a A of a under A in Z 4 produces the curvatures of all 4-tuples of mutually tangent circles in the packing determined by a. (C) Topological monodromy often produces thin groups. E.g. 1: Consider the family of hyperelliptic curves C t : y 2 =(x a 1 )(x a 2 ) (x a r )(x t). Here a 1,...,a r are distinct in C, t varies over S = C \{a 1,...,a r }. Fix a base point t 0, H 1 (C t0 ) = Z 2g,whereg = genus(c t0 ).

10 Traverse the closed loop γ and follow a cycle β in H 1 (C t0 ) gives M(γ)β H 1 (C t0 ), representation M : π 1 (S, t 0 ) Sp(2g, Z) monodromy Sp : X t JX = J, J = 0 I I 0. Image(M) iszariskidenseinsp(2g). K. Yu (1990 s): M(π 1 (S)) is finite index in Sp(2g, Z), not thin.

11 However, the family C t : y 5 = x 3 (1 x) 3 (1 tx) 2 corresponds to a nonarithmetic triangle group (Paula Cohen-Wolfart). Zcl(M(π 1 (S))) = H Sp(2g). H is a Hilbert modular subgroup. M(π 1 (S)) is thin (in H(Z)). (D) Veech or Teichmuller curves in M g yield thin monodromy. (E) Do Calabi Yau and Dwork families yield thin monodromy? y y y 3 3 =3ty 4 y 5 y 6 y y y 3 6 =3ty 1 y 2 y 3. (F) Covers of hyperbolic 3-manifolds with large Heegaard genus are given by thin groups (Lackenby, Long Lubotzky Reid).

12 Matthews Weisfeiler Vaeserstein Strong approximation holds for thin groups: Theorem Let Γ SL n (Z) be Zariski dense in SL n. There is a finite set S of primes p 1,...,p v depending on Γ such that for (q, S) =1, Γ SL n (Z/qZ) is onto. Similarly for other simple, simply-connected G s in place of SL n. New treatments: Nori, Larsen Pvik. As for expansion, the familiar number theoretic methods don t work when Vol(Γ \ G(R)) =. However, a combinatorial method going back to S. Xu 1990 s does when combined with many new ideas.

13 (1) Bourgain Gamburd S. general set-up and proof for G = SL 2 ( ). (2) Proof in (1) depends on Helfgott s combinatorial A.A.A. theorem (nonabelian sum-product theorem) for SL 2 (F p ). (3) (2) is generalized to Chevelly groups G(F p ) by Pyber Szabo, Breuillard Green Tao (2010). (4) P. Varjú extends (1) to G = SL n (2010). (5) A. Salehi Varjú prove the most general expander property (2010).

14 Theorem (Super-strong Approximation (Salehi Varjú 2011)) Let Γ GL n (Q) be finitely generated with generating set S. Then the congruence graphs (π q (Γ), S) for q square-free, q prime to a fixed set of primes (depending on Γ) isanexpanderfamilyiff G, the connected component of G = Zcl(Γ), isperfect(g=[g, G]). (effective) This and its earlier versions is at the heart of many diophantine applications. We discuss the affine sieve which is an extension of the Brun Sieve to orbits of affine linear actions.

15 Search for Primes 1-dimension: Z, f Z[x]. Are there infinitely many x such that f (x) isprime? (I) f (x) =x yes. (II) f (x) =ax + b yes if (a, b) = 1, otherwise no (Dirichlet). (III) f (x) =x (Euler conjectured yes). (IV) f (x) =x(x + 2) are there infinitely many x such that f (x) has at most two prime factors? twin prime conjecture. Brun: There are infinitely many x such that f (x) =x(x + 2) has at most 20 prime factors.

16 Saturation Number Let r 0 (Z, f )betheleastr such that the set of x Z which have at most r prime factors is infinite (better for higher dimensions) the least r such that Zcl ({x Z : f (x) has at most r prime factors}) =A 1. Brun: for any f, r 0 (Z, f )isfinite! More generally, let O = a Γ, Γ SL n (Z) be the orbit of a Z n under Γ.

17 Let f Z[x 1,...,x n ]. Set r 0 (O, f )betheleastr (if it exists) such that Zcl ({x O : f (x) has at most r prime factors}) =Zcl(O). Enemy is a torus (for saturation). E.g. O = Γ = {2 m : m Z} GL 2 (Q) a torus. Set F (x) =(x 1)(x 2). Then the standard heuristics suggest that the number of prime factors of (2 m 1)(2 m 2) goes to infinity with m, i.e.r 0 (Γ, F )=.

18 So we must avoid tori that are in the radical of G. The following was conjectured in B G S. Fundamental Theorem of the Affine Sieve (Salehi S. 2011) Let Γ GL n (Z), O = a Γ Z n.ifg= Zcl(Γ) is Levi semisimple (i.e. rad G contains no torus) then for f Z[x 1,...,x n ] with f Zcl(O) 0 (on any compact), r 0 (O, f ) <. Thatis,thereisan r < (effective but not feasible) such that Zcl ({x O : f (x) has at most r prime factors}) =Zcl(O).

19 This applies to integral Apollonian packings. For these and certain f s there are some gems. Theorem (S. 07) There are infinitely many circles with curvature a prime number in any integral Apollonian packing. In fact, there are infinitely many pairs of tangent circles ( twin primes ), both of whose curvatures are prime. In fact, r 0 (O a, x 1 )=1, r 0 (O a, x 1 x 2 )=2.

20 Zaremba s Conjecture: For A large ( 5) and fixed, let D A be the positive integers q such that there is 1 b q 1, (b, q) = 1, with Conjecture b q =[a 1,...,a k ] a j A. D A = N. continued fraction Equivalently, let Γ A be the semi-subgroup of SL 2 (Z) generated by 0 1, 1 a A 1 a b = 1 a 1 1 a 2 1 a k q b q =[a 1,...,a k ].

21 So the conjecture is equivalent to the orbit of (0, 1) under Γ A having second coordinate q for any given q 1. Γ A is thin. This is a local to global question for thin semigroups. Theorem (Bourgain Kontorovich 2011) For A 3000 fixed, D A has density 1, i.e. almost all q in the sense of density are in D A.

22 One of the many new ingredients in the proof of expansion for thin groups is sum product theory from additive combinatorics. Theorem (Bourgain, Nets Katz, Tao) Given ε > 0, thereisδ > 0 such that for p any large prime and A F p with p ε A p 1 ε, A + A + A A A 1+δ.

23 Some references: J. Bourgain, A. Gamburd, and P. Sarnak, Invent. Math. 179 (2010), J. Bourgain and A. Kontorovich, arxiv: E. Breuillard, B. Green, and T. Tao, Linear approximate groups, arxiv: Y. Chen, Y. Yang, and N. Yui, Monodromy of Pichard Fuchs differential equations for Calabi Yau threefolds, arxiv H. Helfgott, Ann. Math. 167 (2008), S. Hoory, N. Linial, and A. Wigderson, Expander graphs and their applications, BAMS 43 (2006), A. Lubotzky, Expander Graphs in Pure and Applied Math. K. A. Mikhailova, Dokl. Akad. Nauk SSSR 119 (1958),

24 L. Pyber and E. Szabo, Growth in finite simple groups of Lie type, arxiv: A. Salehi and P. Sarnak, Affine Sieve, preprint, A. Salehi and P. Varjú, Expansion in perfect groups, preprint, P. Sarnak, Integral Apollonian Packing, Amer. Math. Monthly 118 (2011), P. Varjú, Expansion in SL d (O/I ), arxiv: