Uniform spectral gaps and orthogeodesic counting for strong convergence of geometrically finite Kleinian groups

Spectral gaps of the Laplacian are interesting geometric quantities that have applications in dynamics. In this talk we will study the convergence of small eigenvalues for geometrically finite hyperbolic $n$-manifolds under strong limits. We will apply this together with the exponentially mixing properties of geodesic flow to count uniformly along the sequence the number of orthogeodesics between converging Margulis tubes, which can give uniform control of the size of regions of small injectivity radius. This is joint work with Beibei Liu.