Effective equidistribution of horospherical flows in infinite volume rank one homogeneous spaces

Abstract

We prove effective equidistribution of horospherical flows in SO(n, 1) • /Γ when Γ is either convex cocompact, or is geometrically finite with H n /Γ having all cusps of maximal rank, and the frame flow is exponentially mixing for the Bowen-Margulis-Sullivan measure. We also discuss settings in which such an exponential mixing result is known to hold.

Distribution of orbits of geometrically finite groups acting on null vectors

“…Observe also that in the lattice case, this condition is always satisfied, because Λ(Γ) = ∂(H n ). See [17] for further discussion of this definition. Definition 1.6.…”

“…The assumptions on Γ in Definition 1.6 are to ensure the effective equidistribution theorem in[17, Theorem 1.4] holds (see Theorem 2.12 for a statement of this theorem in this setting). As discussed in[17], this theorem holds whenever all cusps have maximal rank and the frame flow satisfies an explicit exponential mixing statement,[17, Assumption 1.1]. The conditions in Defintion 1.6 include the cases where this exponential mixing is known to hold.…”