Arithmetic Groups

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. The existence of such noncongruence subgroups is reflective of the dual nature of $SL(2,Z)$ as both the mapping class group of a torus, as well as that of a once punctured torus. This coincidence, however innocuous at first sight, leads to quickly diverging theories, the difference being akin to the distinction between surfaces of genus 1 vs genus $\geq 2$ in geometry. In the first part of the talk, I will explain how in the latter perspective, $SL(2,Z)$ actually satisfies a "congruence subgroup property" (in a necessarily different sense!), and how this property leads to a natural procedure that takes as input a topological covering $X$ of a punctured torus $T$, and produces as output a finite index subgroup of $SL(2,Z)$. Moreover, almost every finite index subgroup can be constructed in this way. Geometrically, on input $X/T$, the output is the fundamental group of the moduli stack of covers of punctured elliptic curves which are topologically isomorphic to $X/T$. I will also explain this procedure in terms of group theory, and in terms of Teichmuller dynamics. I will describe some results and natural questions that arise, using the congruence case as an example. While there is a simple algorithm for carrying out the above procedure, it is difficult to predict properties of the output given knowledge of the input. In the second part of the talk, I will describe a new result in this direction. If the input is a cover $X/T$ which is Galois with all ramification indices above the puncture equal to e, then under certain conditions the output will have index divisible by e. Using the character variety for $SL(2)$-representations of $\pi_1(T)$, I will explain how this implies a conjecture of Bourgain, Gamburd, and Sarnak on the arithmetic of the Markoff surface $x^2 + y^2 + z^2 - xyz = 0$. This also leads to an explicit genus formula for a certain family of noncongruence modular curves, and a formula for the Siegel-Veech constants of a certain natural family of square-tiled congruence modular curves.