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

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

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

Received  March 06, 2019 Revised  June 21, 2019 Published  September 2019

Fund Project: HO: Supported in part by NSF Grant #1900101

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 $.

Citation: Minju Lee, Hee Oh. Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $. Journal of Modern Dynamics, 2019, 15: 263-276. doi: 10.3934/jmd.2019021
References:
[1]

Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47. doi: 10.1007/PL00001613. Google Scholar

[2]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces, Ann. of Math., 174 (2011), 1111-1162. doi: 10.4007/annals.2011.174.2.8. Google Scholar

[3]

Y. Benoist and H. Oh, Fuchsian groups and compact hyperbolic surfaces, Enseign. Math., 62 (2016), 189-198. doi: 10.4171/LEM/62-1/2-11. Google Scholar

[4]

Y. Benoist and H. Oh, Geodesic planes in geometrically finite acylindrical 3-manifolds, preprint, arXiv: 1802.04423.Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit Distributions of Orbits of Unipotent Flows and Values of Quadratic Forms, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. Google Scholar

[6]

P. Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2), 95 (1972), 492–510. doi: 10.2307/1970869. Google Scholar

[7]

L. Flaminio and R. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner's rigidity theorem, Invent. Math., 99 (1990), 601-626. doi: 10.1007/BF01234433. Google Scholar

[8]

M. Lee and H. Oh, Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends, preprint, arXiv: 1902.06621.Google Scholar

[9]

G. A. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174. Google Scholar

[10]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565. Google Scholar

[11]

C. McMullenA. Mohammadi and H. Oh, Geodesic planes in hyperbolic $3$-manifolds, Invent. Math., 209 (2017), 425-461. doi: 10.1007/s00222-016-0711-3. Google Scholar

[12]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in the convex core of an acylindrical 3-manifold, preprint, arXiv: 1802.03853.Google Scholar

[13]

S. Mozes and N. A. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159. doi: 10.1017/S0143385700008282. Google Scholar

[14]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545–607. doi: 10.2307/2944357. Google Scholar

[15]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8. Google Scholar

[16]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574. Google Scholar

[17]

N. A. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, in Lie Groups and Ergodic Theory, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998,229–271. Google Scholar

[18]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 281-304. doi: 10.1215/00127094-2009-027. Google Scholar

[19]

D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math., 210 (2015), 467-507. doi: 10.1007/s11856-015-1258-5. Google Scholar

show all references

References:
[1]

Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47. doi: 10.1007/PL00001613. Google Scholar

[2]

Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces, Ann. of Math., 174 (2011), 1111-1162. doi: 10.4007/annals.2011.174.2.8. Google Scholar

[3]

Y. Benoist and H. Oh, Fuchsian groups and compact hyperbolic surfaces, Enseign. Math., 62 (2016), 189-198. doi: 10.4171/LEM/62-1/2-11. Google Scholar

[4]

Y. Benoist and H. Oh, Geodesic planes in geometrically finite acylindrical 3-manifolds, preprint, arXiv: 1802.04423.Google Scholar

[5]

S. G. Dani and G. A. Margulis, Limit Distributions of Orbits of Unipotent Flows and Values of Quadratic Forms, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. Google Scholar

[6]

P. Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2), 95 (1972), 492–510. doi: 10.2307/1970869. Google Scholar

[7]

L. Flaminio and R. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner's rigidity theorem, Invent. Math., 99 (1990), 601-626. doi: 10.1007/BF01234433. Google Scholar

[8]

M. Lee and H. Oh, Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends, preprint, arXiv: 1902.06621.Google Scholar

[9]

G. A. Margulis, Problems and conjectures in rigidity theory, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000,161–174. Google Scholar

[10]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565. Google Scholar

[11]

C. McMullenA. Mohammadi and H. Oh, Geodesic planes in hyperbolic $3$-manifolds, Invent. Math., 209 (2017), 425-461. doi: 10.1007/s00222-016-0711-3. Google Scholar

[12]

C. McMullen, A. Mohammadi and H. Oh, Geodesic planes in the convex core of an acylindrical 3-manifold, preprint, arXiv: 1802.03853.Google Scholar

[13]

S. Mozes and N. A. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159. doi: 10.1017/S0143385700008282. Google Scholar

[14]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545–607. doi: 10.2307/2944357. Google Scholar

[15]

M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J., 63 (1991), 235-280. doi: 10.1215/S0012-7094-91-06311-8. Google Scholar

[16]

N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann., 289 (1991), 315-334. doi: 10.1007/BF01446574. Google Scholar

[17]

N. A. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, in Lie Groups and Ergodic Theory, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998,229–271. Google Scholar

[18]

N. A. Shah, Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds, Duke Math. J., 148 (2009), 281-304. doi: 10.1215/00127094-2009-027. Google Scholar

[19]

D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math., 210 (2015), 467-507. doi: 10.1007/s11856-015-1258-5. Google Scholar

[1]

Marc Peigné. On some exotic Schottky groups. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 559-579. doi: 10.3934/dcds.2011.31.559

[2]

Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, Ⅱ. Journal of Modern Dynamics, 2017, 11: 1-16. doi: 10.3934/jmd.2017001

[3]

Pilar Bayer, Dionís Remón. A reduction point algorithm for cocompact Fuchsian groups and applications. Advances in Mathematics of Communications, 2014, 8 (2) : 223-239. doi: 10.3934/amc.2014.8.223

[4]

Xin Zhang. The gap distribution of directions in some Schottky groups. Journal of Modern Dynamics, 2017, 11: 477-499. doi: 10.3934/jmd.2017019

[5]

Jayadev S. Athreya, Gregory A. Margulis. Logarithm laws for unipotent flows, I. Journal of Modern Dynamics, 2009, 3 (3) : 359-378. doi: 10.3934/jmd.2009.3.359

[6]

Shucheng Yu. Logarithm laws for unipotent flows on hyperbolic manifolds. Journal of Modern Dynamics, 2017, 11: 447-476. doi: 10.3934/jmd.2017018

[7]

L. Bakker. Semiconjugacy of quasiperiodic flows and finite index subgroups of multiplier groups. Conference Publications, 2005, 2005 (Special) : 60-69. doi: 10.3934/proc.2005.2005.60

[8]

Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25

[9]

S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86.

[10]

François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39

[11]

Darryl D. Holm, Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics, 2009, 1 (2) : 181-208. doi: 10.3934/jgm.2009.1.181

[12]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91

[13]

Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477

[14]

Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113

[15]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162

[16]

Douglas A. Leonard. A weighted module view of integral closures of affine domains of type I. Advances in Mathematics of Communications, 2009, 3 (1) : 1-11. doi: 10.3934/amc.2009.3.1

[17]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177

[18]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221

[19]

Cheng Zheng. Sparse equidistribution of unipotent orbits in finite-volume quotients of $\text{PSL}(2,\mathbb R)$. Journal of Modern Dynamics, 2016, 10: 1-21. doi: 10.3934/jmd.2016.10.1

[20]

Thierry Goudon, Martin Parisot. Non--local macroscopic models based on Gaussian closures for the Spizer-Härm regime. Kinetic & Related Models, 2011, 4 (3) : 735-766. doi: 10.3934/krm.2011.4.735

2018 Impact Factor: 0.295

Article outline

[Back to Top]