Horospherically invariant measures and finitely generated Kleinian groups

Abstract

Let \begin{document}$ \Gamma < {\rm{PSL}}_2( \mathbb{C}) $\end{document} be a Zariski dense finitely generated Kleinian group. We show all Radon measures on \begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document} which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [18] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [2] and Calegari-Gabai [10].