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.