Horospherically invariant measures and finitely generated Kleinian groups

TL;DR

This paper classifies ergodic horospherically invariant measures on the quotient space of a Zariski dense finitely generated Kleinian group, revealing they are either supported on a single orbit or quasi-invariant under the geodesic flow.

Contribution

It establishes a classification of invariant measures for Kleinian groups using advanced techniques from dynamics and 3-manifold theory, including the Tameness Theorem.

Findings

Measures are supported on a single closed horospherical orbit or are quasi-invariant.

The classification leverages recent results in 3-manifold topology.

The approach combines dynamics with geometric topology.

Abstract

Let $ \Gamma < PSL_2(\mathbb{C}) $ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on $ PSL_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.