Workshop on Thin Groups and Super Approximation

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

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...

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...

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.

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 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...

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)...

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...

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.