## Workshop on Thin Groups and Super Approximation

Please see schedule

March 28, 2016

Please see schedule

March 28, 2016

Alireza Salehi Golsefidy

UCSD

March 29, 2016

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.

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.

Peter Sarnak

Institute for Advanced Study

March 29, 2016

Alex Kontorovich

Rutgers/IAS

March 29, 2016

Alan Reid

University of Texas

March 29, 2016

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.

as a method to certify that a group is thin. This involves a mix of theory and computation.

Elena Fuchs

University of Illinois at Urbana–Champaign

March 30, 2016

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

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

Alex Lubotzky

The Hebrew University of Jerusalem

March 30, 2016

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

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

Jean Bourgain

Institute for Advanced Study

March 31, 2016

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.

triples together with an extension of Zagier’s counting result, we show that most Markoff

numbers are composite.

T. N. Venkataramana

Tata Institute of Fundamental Research

March 31, 2016

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.

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.

Darren Long

University of California, Santa Barbara

March 31, 2016

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.

as a method to certify that a group is thin. This involves a mix of theory and computation.