Horospherically invariant measures and finitely generated Kleinian groups

Abstract

Let $\Gamma <PS{L}_2(\mathbb{C})$ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on $PS{L}_2(\mathbb{C})/\Gamma$ 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 together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol and Calegari-Gabai.

16 pages, 1 figure, results unchanged