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

Abstract

AbstractLet $$\Gamma $$ Γ be a geometrically finite discrete subgroup in $${\text {SO}}(d+1,1)^{\circ }$$ SO ( d + 1 , 1 ) ∘ with parabolic elements. We establish exponential mixing of the geodesic flow on the unit tangent bundle $${\text {T}}^1(\Gamma \backslash {\mathbb {H}}^{d+1})$$ T 1 ( Γ \ H d + 1 ) with respect to the Bowen–Margulis–Sullivan measure, which is the unique probability measure on $${\text {T}}^1(\Gamma \backslash {\mathbb {H}}^{d+1})$$ T 1 ( Γ \ H d + 1 ) with maximal entropy. As an application, we obtain a resonance-free region for the resolvent of the Laplacian on $$\Gamma \backslash {\mathbb {H}}^{d+1}$$ Γ \ H d + 1 . Our approach is to construct a coding for the geodesic flow and then prove a Dolgopyat-type spectral estimate for the corresponding transfer operator.