The topic of my talk is what the possibilities are for the closure of a totally geodesic immersion of a hyperbolic plane into a complete hyperbolic 3 manifold.  For finite volume hyperbolic 3 manifolds, work of Shah and Ratner shows that strong rigidity properties hold: any such immersed plane is either closed or dense.  I will describe recent joint work with Curt McMullen and Amir Mohammadi  which shows some of this rigidity persists in some cases of infinite volume hyperbolic 3-manifolds

For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss  exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for  the associated Riemannian manifold.  We will also discuss a uniform  exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.