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 and 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.

[2]

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

[3]

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

[4]

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

[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.

[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. 

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[4]

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

[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.

[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. 

[7]

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

[8]

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

[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.

[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.

[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.

[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.

[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.

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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