February  2019, 39(2): 881-903. doi: 10.3934/dcds.2019037

Geodesic planes in geometrically finite manifolds

Department of Mathematics, Ohio State University, 231 W 18th Ave., Columbus, OH 43210, USA

Received  January 2018 Revised  July 2018 Published  November 2018

We study the problem of rigidity of closures of totally geodesic plane immersions in geometrically finite manifolds containing rank 1 cusps. We show that the key notion of K-thick recurrence of horocycles fails generically in this setting. This property played a key role in the recent breakthroughs of McMullen, Mohammadi and Oh. Nonetheless, in the setting of geometrically finite groups whose limit sets are circle packings, we derive 2 density criteria for non-closed geodesic plane immersions, and show that closed immersions give rise to surfaces with finitely generated fundamental groups. We also obtain results on the existence and isolation of proper closed immersions of elementary surfaces.

Citation: Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037
References:
[1]

R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58158-8.  Google Scholar

[2]

B. H. Bowditch, Geometrical finiteness for hyperbolic groups, Journal of Functional Analysis, 113 (1993), 245-317.  doi: 10.1006/jfan.1993.1052.  Google Scholar

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F. Dal'bo, Topologie du feuilletage fortement stable, Annales de l'institut Fourier, 50 (2000), 981-993.   Google Scholar

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R. L. GrahamJ. C. LagariasC. L. MallowsA. R. Wilks and C. H. Yan, Apollonian circle packings: Number theory, J. Number Theory, 100 (2003), 1-45.  doi: 10.1016/S0022-314X(03)00015-5.  Google Scholar

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L. KeenB. Maskit and C. Series, Geometric finiteness and uniqueness for kleinian groups with circle packing limit sets, J. Reine Angew. Math., 436 (1993), 209-219.   Google Scholar

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G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Banach Center Publications, 23 (1989), 399-409.   Google Scholar

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F. Maucourant and B. Schapira, On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds, ArXiv e-prints, February 2017. Google Scholar

[9]

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

[10]

C. T McMullenA. Mohammadi and H. Oh, Horocycles in hyperbolic 3-manifolds, Geometric and Functional Analysis, 26 (2016), 961-973.  doi: 10.1007/s00039-016-0373-8.  Google Scholar

[11]

H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of kleinian groups, Inventiones Mathematicae, 187 (2012), 1-35.  doi: 10.1007/s00222-011-0326-7.  Google Scholar

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

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N.Shah, Closures of totally geodesic immersions in manifolds of constant negative curvature,Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Scientific, (1991), 718-732.  Google Scholar

show all references

References:
[1]

R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitext. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58158-8.  Google Scholar

[2]

B. H. Bowditch, Geometrical finiteness for hyperbolic groups, Journal of Functional Analysis, 113 (1993), 245-317.  doi: 10.1006/jfan.1993.1052.  Google Scholar

[3]

F. Dal'bo, Topologie du feuilletage fortement stable, Annales de l'institut Fourier, 50 (2000), 981-993.   Google Scholar

[4]

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

[5]

R. L. GrahamJ. C. LagariasC. L. MallowsA. R. Wilks and C. H. Yan, Apollonian circle packings: Number theory, J. Number Theory, 100 (2003), 1-45.  doi: 10.1016/S0022-314X(03)00015-5.  Google Scholar

[6]

L. KeenB. Maskit and C. Series, Geometric finiteness and uniqueness for kleinian groups with circle packing limit sets, J. Reine Angew. Math., 436 (1993), 209-219.   Google Scholar

[7]

G. A. Margulis, Indefinite quadratic forms and unipotent flows on homogeneous spaces, Banach Center Publications, 23 (1989), 399-409.   Google Scholar

[8]

F. Maucourant and B. Schapira, On topological and measurable dynamics of unipotent frame flows for hyperbolic manifolds, ArXiv e-prints, February 2017. Google Scholar

[9]

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

[10]

C. T McMullenA. Mohammadi and H. Oh, Horocycles in hyperbolic 3-manifolds, Geometric and Functional Analysis, 26 (2016), 961-973.  doi: 10.1007/s00039-016-0373-8.  Google Scholar

[11]

H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of kleinian groups, Inventiones Mathematicae, 187 (2012), 1-35.  doi: 10.1007/s00222-011-0326-7.  Google Scholar

[12]

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

[13]

N.Shah, Closures of totally geodesic immersions in manifolds of constant negative curvature,Group Theory from a Geometrical Viewpoint (Trieste, 1990), World Scientific, (1991), 718-732.  Google Scholar

Figure 1.  Apollonian circle packing (solid). Inversions through dual circles (dashed) generate a geometrically finite group containing rank-$1$ parabolic subgroups
Figure 2.  Proof of Lemma 3.2
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