Horospherically invariant measures and finitely generated Kleinian groups

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Gamma &lt; {\rm{PSL}}_2( \mathbb{C}) $\end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document}</tex-math></inline-formula> 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 [<xref ref-type="bibr" rid="b18">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type="bibr" rid="b2">2</xref>] and Calegari-Gabai [<xref ref-type="bibr" rid="b10">10</xref>].</p>