In an effort to blog a bit more, and to spur myself to work harder, I’m going to try to blog more about the work I’m doing this summer.
Briefly, I’m exploring properties of subsets of the modular group, defined as the set of 2×2 matrices with determinant 1. I work with an eye toward the problem of generating congruence subgroups, which is not adequately solved. It is possible to check whether a given group is a subgroup, but it has shown to be difficult to work in the opposite direction.
This week I’m delving into Topology. Any modular subgroup can be described as a polygon on the hyperbolic plane, which in turn can be turned into a Riemann surface by identifying corresponding edges (see for example Wohlfahrt). Of particular interest is the nature of the covering of that surface. Read beneath the fold for diffeomorphisms.