Abstract: Let H^n be the hyperbolic n-space and D be a geometrically finite discrete subgroup in Isom_+(H^n) with cusps. In the joint work with Wenyu Pan, we establish exponential mixing of the geodesic flow over the unit tangent bundle T^1(D \ H^n).
Previously, such results were proved by Stoyanov for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for D with large critical exponent. We obtain our result by constructing a nice coding for the geodesic flow and then prove a Dolgopyat-like spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding, which is partly inspired by the works of Lai-Sang Young and Burns-Masur-Matheus-Wilkinson. I will also discuss the application of obtaining a resonance-free region for the resolvent on D\H^n.
