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Prescribing the behaviour of geodesics in negative curvature. (English) Zbl 1191.53026

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold \(M\), such as balls, horoballs, tubular neighbourhoods of totally geodesic submanifolds, etc., the aim of this paper is to construct geodesic rays or lines in \(M\) which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much into) them. Several applications are given, including a definite improvement of the unclouding problem of an earlier paper of the authors [Geom. Funct. Anal. 15, No. 2, 491–533 (2005; Zbl 1082.53035)], the prescription of heights of geodesic lines in such an \(M\) of finite volume, or of spiraling times around a closed geodesic in a closed such \(M\). They also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property. The work has deep connections with Diophantine approximation problems.

MSC:

53C22 Geodesics in global differential geometry
52A55 Spherical and hyperbolic convexity
53D25 Geodesic flows in symplectic geometry and contact geometry
11J06 Markov and Lagrange spectra and generalizations

Citations:

Zbl 1082.53035

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[1] W Abikoff, B Maskit, Geometric decompositions of Kleinian groups, Amer. J. Math. 99 (1977) 687 · Zbl 0374.30018
[2] L V Ahlfors, Möbius transformations and Clifford numbers (editors I Chavel, H M Farkas), Springer (1985) 65 · Zbl 0569.30040
[3] C S Aravinda, E Leuzinger, Bounded geodesics in rank-\(1\) locally symmetric spaces, Ergodic Theory Dynam. Systems 15 (1995) 813 · Zbl 0835.58026
[4] H Aslaksen, Quaternionic determinants, Math. Intelligencer 18 (1996) 57 · Zbl 0881.15007
[5] A F Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer (1983) · Zbl 0528.30001
[6] A Borel, Linear algebraic groups, Amer. Math. Soc. (1966) 3 · Zbl 0205.50503
[7] A Borel, Reduction theory for arithmetic groups, Amer. Math. Soc. (1966) 20 · Zbl 0213.47201
[8] A Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. \((2)\) 75 (1962) 485 · Zbl 0107.14804
[9] M Bourdon, Sur le birapport au bord des \(\mathrm{CAT}(-1)\)-espaces, Inst. Hautes Études Sci. Publ. Math. 83 (1996) 95 · Zbl 0883.53047
[10] B H Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995) 229 · Zbl 0877.57018
[11] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[12] P Buser, H Karcher, Gromov’s almost flat manifolds, Astérisque 81, Société Mathématique de France (1981) 148 · Zbl 0459.53031
[13] S Buyalo, V Schroeder, M Walz, Geodesics avoiding open subsets in surfaces of negative curvature, Ergodic Theory Dynam. Systems 20 (2000) 991 · Zbl 1039.53039
[14] D A Cox, Primes of the form \(x^2 + ny^2\), Fermat, class field theory and complex multiplication, Wiley-Interscience, John Wiley & Sons (1989) · Zbl 0701.11001
[15] T W Cusick, M E Flahive, The Markoff and Lagrange spectra, Math. Surveys and Monogr. 30, Amer. Math. Soc. (1989) · Zbl 0685.10023
[16] S G Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986) 636 · Zbl 0627.22013
[17] J Dieudonné, Les déterminants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943) 27 · Zbl 0028.33904
[18] J Elstrodt, F Grunewald, J Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory, Springer Monogr. in Math., Springer (1998) · Zbl 0888.11001
[19] J M Feustel, Über die Spitzen von Modulflächen zur zweidimensionalen komplexen Einheitskugel, Preprint Series 13 Akad. Wiss. DDR, ZIMM, Berlin 14 (1977)
[20] L R Ford, Rational approximations to irrational complex numbers, Trans. Amer. Math. Soc. 19 (1918) 1 · JFM 46.0275.04
[21] G A Freiman, Diophantine approximations and the geometry of numbers (Markov’s problem), Kalinin. Gosudarstv. Univ., Kalinin (1975) 144 · Zbl 0347.10025
[22] É Ghys, P de la Harpe, editors, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Math. 83, Birkhäuser (1990) · Zbl 0731.20025
[23] W M Goldman, Complex hyperbolic geometry, Oxford Math. Monogr., Oxford Univ. Press (1999) · Zbl 0939.32024
[24] M Hall Jr., On the sum and product of continued fractions, Ann. of Math. \((2)\) 48 (1947) 966 · Zbl 0030.02201
[25] M Hall Jr., The Markoff spectrum, Acta Arith. 18 (1971) 387 · Zbl 0224.10023
[26] U Hamenstädt, A new description of the Bowen-Margulis measure, Ergodic Theory Dynam. Systems 9 (1989) 455 · Zbl 0722.58029
[27] Y Hellegouarch, Quaternionic homographies: application to Ford hyperspheres, C. R. Math. Rep. Acad. Sci. Canada 11 (1989) 171 · Zbl 0711.51004
[28] S Hersonsky, F Paulin, On the volumes of complex hyperbolic manifolds, Duke Math. J. 84 (1996) 719 · Zbl 0866.53036
[29] S Hersonsky, F Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997) 349 · Zbl 0908.57009
[30] S Hersonsky, F Paulin, Diophantine approximation for negatively curved manifolds, Math. Z. 241 (2002) 181 · Zbl 1037.53020
[31] S Hersonsky, F Paulin, Diophantine approximation in negatively curved manifolds and in the Heisenberg group (editors M Burger, A Iozzi), Springer (2002) 203 · Zbl 1064.11057
[32] S Hersonsky, F Paulin, Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions, Ergodic Theory Dynam. Systems 24 (2004) 803 · Zbl 1059.37022
[33] S Hersonsky, F Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds, to appear in J. Diff. Geom. · Zbl 1229.53050
[34] R Hill, S L Velani, The Jarník-Besicovitch theorem for geometrically finite Kleinian groups, Proc. London Math. Soc. \((3)\) 77 (1998) 524 · Zbl 0924.11063
[35] R P Holzapfel, Arithmetische Kugelquotientenflächen I/II, Seminarberichte 14, Humboldt Univ. Sekt. Math. (1978)
[36] R P Holzapfel, Ball and surface arithmetics, Aspects of Math., E29, Friedr. Vieweg & Sohn (1998) · Zbl 0980.14026
[37] S Kamiya, On discrete subgroups of \(\mathrm{PU}(1,2;\mathbfC)\) with Heisenberg translations, J. London Math. Soc. \((2)\) 62 (2000) 827 · Zbl 1011.22006
[38] R Kellerhals, Quaternions and some global properties of hyperbolic \(5\)-manifolds, Canad. J. Math. 55 (2003) 1080 · Zbl 1054.57019
[39] D Y Kleinbock, G A Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999) 451 · Zbl 0934.22016
[40] D Y Kleinbock, B Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not. 30 (2004) 1551 · Zbl 1075.37008
[41] A Korányi, H M Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985) 309 · Zbl 0567.30017
[42] C Maclachlan, A W Reid, The arithmetic of hyperbolic \(3\)-manifolds, Graduate Texts in Math. 219, Springer (2003) · Zbl 1025.57001
[43] C Maclachlan, P L Waterman, N Wielenberg, Higher-dimensional analogues of the modular and Picard groups, Trans. Amer. Math. Soc. 312 (1989) 739 · Zbl 0673.20022
[44] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Math. Monogr., Oxford Univ. Press (1998) · Zbl 0892.30035
[45] J P Otal, Sur la géometrie symplectique de l’espace des géodésiques d’une variété à courbure négative, Rev. Mat. Iberoamericana 8 (1992) 441 · Zbl 0777.53042
[46] J R Parker, Shimizu’s lemma for complex hyperbolic space, Internat. J. Math. 3 (1992) 291 · Zbl 0761.32014
[47] J Parkkonen, F Paulin, Unclouding the sky of negatively curved manifolds, Geom. Funct. Anal. 15 (2005) 491 · Zbl 1082.53035
[48] J Parkkonen, F Paulin, Sur les rayons de Hall en approximation diophantienne, C. R. Math. Acad. Sci. Paris 344 (2007) 611 · Zbl 1151.11030
[49] J Parkkonen, F Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds, Preprint (2008) · Zbl 1228.53055
[50] J Parkkonen, F Paulin, On strictly convex subsets in negatively curved manifolds, in preparation · Zbl 1259.53018
[51] F Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. \((2)\) 54 (1996) 50 · Zbl 0854.20050
[52] G Poitou, Sur l’approximation des nombres complexes par les nombres des corps imaginaires quadratiques dénués d’idéaux non principaux, particulièrement lorsque vaut l’algorithme d’Euclide, Ann. Sci. Ecole Norm. Sup. \((3)\) 70 (1953) 199 · Zbl 0053.02802
[53] A L Schmidt, Farey simplices in the space of quaternions, Math. Scand. 24 (1969) 31 · Zbl 0186.09302
[54] A L Schmidt, On the approximation of quaternions, Math. Scand. 34 (1974) 184 · Zbl 0291.10026
[55] T A Schmidt, M Sheingorn, Riemann surfaces have Hall rays at each cusp, Illinois J. Math. 41 (1997) 378 · Zbl 0877.30019
[56] V Schroeder, Bounded geodesics in manifolds of negative curvature, Math. Z. 235 (2000) 817 · Zbl 0990.53038
[57] C Series, The modular surface and continued fractions, J. London Math. Soc. \((2)\) 31 (1985) 69 · Zbl 0545.30001
[58] N J A Sloane, editor, The online encyclopedia of integer sequences · Zbl 1274.11001
[59] B Stratmann, The Hausdorff dimension of bounded geodesics on geometrically finite manifolds, Ergodic Theory Dynam. Systems 17 (1997) 227 · Zbl 0869.53029
[60] D Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982) 215 · Zbl 0517.58028
[61] M F Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer (1980) · Zbl 0422.12008
[62] R Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ. Hamburg 45 (1976) 263 · Zbl 0332.53026
[63] T Zink, Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen, Math. Nachr. 89 (1979) 315 · Zbl 0424.10020
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