Superstrong approximation

Abstract: This book constitutes articles by participants at the workshop ‘Thin groups and superstrong approximation’, held at MSRI in 2012. Many of the contributors are leading mathematicians, and the topic is one of great current interest with connections to many parts of mathematics: number theory, Lie theory, additive combinatorics and ergodic theory, to name a few. Neither of the technical terms in the title are particularly standard yet, though they should become so in due course. Let G be a connected semisimple algebraic group. For the purposes of reading this review, the reader may assume that G = SLn. A thin group is a discrete subgroup Γ of G(R) that is Zariskidense in G (that is, not contained in any proper subvariety) and yet has infinite covolume in G(R). One may contrast this with the notion of a lattice, which has the same properties except for having finite covolume in G(R). The articles of Fuchs, and of Long and Reid, give a variety of constructions and instances of such groups. In order to understand what is meant by superstrong approximation, it is natural to first discuss strong approximation. A classical setting for this would involve looking at Γ = SLn(Z), which is a lattice in SLn(R). Strong approximation in this setting refers to the phenomenon that the reduction map π : SLn(Z) → SLn(Z/qZ) is surjective, for all q 1. Superstrong approximation refers to the fact that, roughly speaking, this map is very efficiently surjective. Indeed, given a finite set S of generators for SLn(Z), the reduction π(S) generates SLn(Z/qZ) extremely rapidly and uniformly; in fact, the random walk on generating set π(S) becomes highly equidistributed in time On(log q). (This property is known as expansion.) Superstrong approximation for lattices is a fairly classical topic with contributions by Selberg, Burger, Sarnak, Clozel and many others. That the same phenomenon also exists for thin groups is more recent, and much progress has been possible in recent years due to advances in additive combinatorics (discussed in Breuillard’s article, as well as in that by Pyber and Szabó) and the introduction of a powerful technique by Bourgain and Gamburd. There is now an extremely general theorem in this direction, due to Salehi and Varjú (a precise statement may be found in the articles of Bourgain, of Salehi and of Sarnak in the present volume). Applications in number theory are one of the key justifications for studying thin groups and their approximation properties. A beautiful one is discussed in the articles by Bourgain and Kontorovich, reporting on joint work of the two of them. Given a positive integer A > 1, let DA be the set of all d ∈ N for which at least one of the fractions b/d, hcf(b, d) = 1, has all of the partial quotients in its continued fraction expansion bounded by A. A notorious conjecture of Zaremba asserts that in fact DA = N if A is sufficiently large—perhaps even A = 5 will do. While this problem remains open, Bourgain and Kontorovich have proven that DA has density 1 for A 50. The connection with thin groups comes from the observation that DA is precisely the set of lower right entries of the semigroup ΓA generated by the matrices ( 0 1 1 a ), a A. (As A becomes larger, ΓA remains thin but ‘not too thin’ in the sense that its critical exponent tends to 1.) To explain anything more about this application (including the link with superstrong

Abstract: 1. Some Diophantine applications of the theory of group expansion Jean Bourgain 2. A brief introduction to approximate groups Emmanuel Breuillard 3. Superstrong approximation for monodromy groups Jordan S. Ellenberg 4. The ubiquity of thin groups Elena Fuchs 5. The orbital circle method Alex V. Kontorovich 6. Sieve in discrete groups, especially sparse Emmanuel Kowalski 7. How random are word maps? Michael Larsen 8. Constructing thin groups Darren Long and Alan W. Reid 9. On ergodic properties of the Burger-Roblin measure Amir Mohammadi 10. Harmonic analysis, ergodic theory and counting for thin groups Hee Oh 11. Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces Gopal Prasad and Andrei Rapinchuk 12. Growth in linear groups Laszlo Pyber and Endre Szabo 13. On strong approximation for algebraic groups Andrei Rapinchuk 14. Generic phenomena in groups: some answers and many questions Igor Rivin 15. Affine sieve and expanders Alireza Salehi Golsefidy 16. Growth in linear groups Peter Sarnak.

Abstract: These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group theory, gonality of curves and Heegaard genus of hyperbolic three manifolds, are given. We also discuss the ubiquity of thin matrix groups in various contexts, and in particular that of monodromy groups.