Simply transitive geodesics and omnipotence of lattices in PSL$(2,\mathbb{C})$

Abstract

We show that the isometry group of a finite-volume hyperbolic 3-manifold acts simply transitively on many of its closed geodesics. Combining this observation with the Virtual Special Theorems of the first author and Wise, we show that every non-arithmetic lattice in PSL$(2,\mathbb{C})$ is the full group of orientation-preserving isometries for some other lattice and that the orientation-preserving isometry group of a finite-volume hyperbolic 3-manifold acts non-trivially on the homology of some finite-sheeted cover.