Exponential mixing of geodesic flows for geometrically finite hyperbolic manifolds with cusps

Group Actions and Dynamics
Event time: 
Monday, September 21, 2020 - 4:00pm
Wenyu Pan
Speaker affiliation: 
University of Chicago
Event description: 

Let $\mathbb{H}^n$ be the hyperbolic $n$-space and $\Gamma$ be a geometrically finite discrete subgroup in $\operatorname{Isom}_{+}(\mathbb{H}^n)$ with parabolic elements. We establish exponential mixing of the geodesic flow over the unit tangent bundle $T^1(\Gamma\backslash \mathbb{H}^n)$ with respect to the Bowen-Margulis-Sullivan measure. Previously, such results were proved by Naud, Stoyanov and Sarkar-Winter for convex cocompact discrete subgroups and Mohammadi-Oh and Edwards-Oh for $\Gamma$ with large critical exponent. Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator. In the talk, I am planning to explain the construction of the coding. I will also discuss the application of obtaining a resonance-free region for the resolvent on $\Gamma\backslash \mathbb{H}^n$. This is a joint work with Jialun Li.