Exact curve counting on the once-punctured torus

Two closed curves on a surface are said to have the same topological type if they lie in the same mapping class group orbit. Mirzakhani first showed that, on a hyperbolic surface, the number of closed geodesics of a given topological type of length at most L is asymptotically proportional to L^{6g-6}. In this talk, we will focus on the case of the once-punctured torus. Rather than using hyperbolic length, we will count curves according to their word length. In this setting, it turns out that exact counting formulas can be obtained. This is joint work with Filippo Baroni and David Fisac.