Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $

Minju Lee; Hee Oh

doi:10.3934/jmd.2019021

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2019, Volume 15: 263-276. Doi: 10.3934/jmd.2019021

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Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $

Minju Lee and
Hee Oh

Department of Mathematics, Yale University, New Haven, CT 06520, USA

Received: March 05, 2019

Revised: June 20, 2019

Published: September 2019

HO: Supported in part by NSF Grant #1900101

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Abstract

We present a new proof of the following theorem of Benoist-Quint: Let $ G: = \operatorname{SO}^\circ(d, 1) $, $ d\ge 2 $ and $ \Delta<G $ a cocompact lattice. Any orbit of a Zariski dense subgroup $ \Gamma $ of $ G $ is either finite or dense in $ \Delta \backslash G $. While Benoist and Quint's proof is based on the classification of stationary measures, our proof is topological, using ideas from the study of dynamics of unipotent flows on the infinite volume homogeneous space $ \Gamma \backslash G $.

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Mathematics Subject Classification: Primary: 37A17; Secondary: 22E40.

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References

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