Let ${\mathbb{H}^n}$ be the hyperbolic 𝑛-space and Γ be a geometrically finite discrete subgroup in Isom$_+$(${\mathbb{H}^n}$) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T$^1$ (Γ \ ${\mathbb{H}^n}$) (resp. the frame bundle F(Γ \ ${\mathbb{H}^n}$)) mixes exponentially. This result has many applications, including spectral theory, prime geodesic theorems, orbit counting, equidistribution, etc.

I will start with a survey of the past results, methods, and related problems on this topic. Along the way, I will present the joint work with Jialun Li, Pratyush Sarkar.