Workshop on Thin Groups and Super Approximation

Abstract: Let Γ be a finitely generated subgroup of a compact group G. 1. I will recall what it means to say the action of Γ on G by left translation has spectral gap, and mention some examples and applications, e.g. Banach-Ruziewicz problem...

Schedule: https://sites.google.com/site/thingroups/schedule

In this talk we introduce the Apollonian group, sometimes coined the “quintessential” thin group, which is the underlying symmetry group of Apollonian circle packings. We review some of...

I will explain the Bourgain-Gamburd method of proving spectral gap. The spirit of this method is behind all the recent results on this subject. Very roughly it says in order to prove spectral gap it is enough to show that a “generic” subset with...

In this talk the general setting of affine sieve will be presented. Next I will explain the Bourgain-Gamburd-Sarnak method on proving affine sieve in the presence of certain spectral gap. Finally I will say how one can prove the fundamental theorem...

In this talked we will explain how the classical large sieve method from number theory can be applied to study properties of subsets of groups which have property-τ . As an application we will show that the set of powers in finitely-generated non-...

In these two talks we will discuss situations in which geometric input can be used as a method to certify that a group is thin. This involves a mix of theory and computation.

Are most finitely generated groups thin or arithmetic? While philosophic in nature, this question is a natural one to ask in light of the recent surge of interest in thin groups in the number -theoretic context. In this talk, we discuss a result...

Non-commutative super approximation and the product replacement algorithm Abstract: Let A be the free abelian group on n generators and C a finite simple abelian group. The action of Aut(A) on E = Epi(A, C) ( = the set of epimorphisms from A to C)...

Monodromy of nFn−1 hypergeometric functions and arithmetic groups I Abstract: We describe results of Levelt and Beukers-Heckman on the explicit computation of monodromy for generalised hypergeometric functions of one variable. We then discuss the...

Using available results on the strong approximation property for the set of Markoff triples together with an extension of Zagier’s counting result, we show that most Markoff numbers are composite.

In these two talks we will discuss situations in which geometric input can be used as a method to certify that a group is thin. This involves a mix of theory and computation.

In this talk we will expalin the main ideas of the proof of the following theorem of Vinberg: Let f be an integral quadratic form of signature (n, 1). If n ≥ 30 then the subgroup of SO(n, 1)(Z) which is generated by all hyperbolic reflections has...

Combinatorics and Geometry to Arithmetic of Circle Packings Abstract: The Koebe-Andreev-Thurston/Schramm theorem assigns a conformally rigid fi- nite circle packing to a convex polyhedron, and then successive inversions yield a conformally rigid...

We describe results of Levelt and Beukers-Heckman on the explicit computation of monodromy for generalised hypergeometric functions of one variable. We then discuss the question of arithmeticity of these monodromy groups and describe various results...