# American Institute of Mathematical Sciences

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. McMullen, A. 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

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##### 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. McMullen, A. 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
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