# Arithmetic Groups ### Applications to modular forms and noncongruence arithmetic groups

Yunqing Tang and Frank Calegari

We explain our proof of the unbounded denominators conjecture. This talk will require the main theorem of the lecture on Nov. 17, 2021, as a “black box” but otherwise be logically independent of that talk. ### Algebraicity/holonomicity theorems

Vesselin Dimitrov and Frank Calegari

Let f=?anxn??[x] be a power series which is also a meromorphic function in some neighborhood of the origin. The subject of the talk will be how certain conditions on f(x) as a meromorphic function actually guarantee that f(x) is an algebraic... ### The congruence subgroup property for SL(2,Z)

Somehow, despite the title, SL(2,Z) is the poster child for arithmetic groups not satisfying the congruence subgroup property, which is to say that it has finite index subgroups which can not be defined by congruence conditions on their coefficients... ### Groups with bounded generation: properties and examples

After surveying some important consequences of the property of bounded generation (BG) dealing with SS-rigidity, the congruence subgroup problem, etc., we will focus on examples of boundedly generated groups. We will prove that every unimodular (n×n... ### First-order rigidity, bi-interpretability, and congruence subgroups

Nir Avni

I'll describe a method for analyzing the first-order theory of an arithmetic group using its congruence quotients. When this method works, it gives a strong form of first-order rigidity together with a complete description of the collection of... ### First order rigidity of high-rank arithmetic groups

The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics.

It includes SL(n,?), for n>2 , SL(n,?[1/p]) for n>1, their finite index subgroups and many more.

A number of...