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.

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[6]

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

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

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