Uniform spectral gap and orthogeodesic counting for strong convergence of Kleinian groups

Abstract We show convergence of small eigenvalues for geometrically finite hyperbolic n -manifolds under strong limits. For a class convergent convex sets in strongly sequence Kleinian groups, we use the spectral gap limit manifold and exponentially mixing property geodesic flow along to find asymptotically uniform counting formulas number orthogeodesics between sets. In particular, this provides (with respect length) converging Margulis tubes, loops based at basepoints, primitive closed geodesics.