Title: The dark side of the modular group SL(2,Z)
Abstract: The group SL(2,Z) has a number of dual natures. In group theory, it is the (special) automorphism group of Z^2, as well as the special outer automorphism group of a free group of rank 2. In surface topology, it is the mapping class group of both a torus as well as a punctured torus. In the theory of arithmetic groups, it is also special in that it has both congruence, and noncongruence subgroups. While the abelian/unpunctured/congruence side is now well understood, and has led to the profoundly successful theory of congruence modular curves and modular forms, the nonabelian/punctured/noncongruence occupies a rather mysterious spot that manages to simultaneously, and just barely, escape the reach of a number of seemingly relevant results. In this talk we will explain some interesting questions and conjectures in the area, and describe some connections with topics such as combinatorial group theory, representation theory, and number theory.