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Finite index subgroups of SL(2,Z) have long been studied from the number theoretic point of view, for example as the symmetry groups underlying modular forms. However, arithmetic problems involving infinite index or thin subgroups have only recently become approachable due to groundbreaking results in the past decade concerning families of expander graphs. Still, this area of number theory is only in its infancy. In this talk, we discuss the history and recent results concerning the dichotomy of thin vs non-thin groups, and present work on the arithmetic of rather general families of thin groups.

A reception in honor of the speaker will be held following the talk!