Horospherically invariant measures and finitely generated Kleinian groups

Or Landesberg

doi:10.3934/jmd.2021012

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2021, Volume 17: 337-352. Doi: 10.3934/jmd.2021012

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Horospherically invariant measures and finitely generated Kleinian groups

Or Landesberg

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem, 9190401, Israel

Received: September 09, 2020

Revised: February 28, 2021

Published: September 2021

This work was supported by ERC 2020 grant HomDyn (grant no. 833423)

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Abstract

Let $ \Gamma < {\rm{PSL}}_2( \mathbb{C}) $ be a Zariski dense finitely generated Kleinian group. We show all Radon measures on $ {\rm{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 [18] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [2] and Calegari-Gabai [10].

Keywords:

Homogeneous flows,
Kleinian groups,
infinite-measure preserving transformations,
horospherical flow,
Tameness Theorem.

Mathematics Subject Classification: Primary: 37A17, 57K32; Secondary: 37A40, 30F40.

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