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

TL;DR

This paper demonstrates that the isometry group of certain hyperbolic 3-manifolds acts simply transitively on many closed geodesics and explores the implications for lattices in PSL(2,C), including their omnipotence and actions on homology.

Contribution

It establishes new properties of isometry groups acting on hyperbolic 3-manifolds and shows non-arithmetic lattices can be realized as full isometry groups of other lattices.

Findings

Isometry groups act simply transitively on many closed geodesics.

Non-arithmetic lattices in PSL(2,C) can be realized as full isometry groups.

The isometry group acts non-trivially on the homology of some finite covers.

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.