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 of affine sieve
by getting such a result in the absence of spectral gap.
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 joint with Rivin
which says that the generic finitely generated subgroup of SLn(Z) is thin in some sense.
The proof of this result hinges on showing that the generic such group is free, from which
thinness follows almost immediately. A key ingredient is a geometric certificate for freeness
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)
satisfies super-approximation (i.e., the induced graph is an expander). We will discuss the
situation when A is replaced the non-abelian free group F and C a non-abelian finite simple
group S. This question is of interest in presentation theory of finite groups as well as for
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
question of arithmeticity of these monodromy groups and describe various results obtained
in recent years towards this question.