Simply transitive geodesics and omnipotence of lattices in PSL(2,C)

In this talk we will discuss how we can combine geodesic counting theorems in hyperbolic manifolds and theorems about controlled quotients of virtually special groups to see that every non-arithmetic lattice in PSL(2,C) is the full (orientation preserving) isometry group of some other lattice, and that every isometry of a finite-volume hyperbolic 3-manifold acts non-trivially on the first homology of some finite-sheeted cover. This is joint work with Ian Agol and Yair Minsky.