Geometry & Topology Volume 14, issue 1 (2010)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Prescribing the behaviour of geodesics in negative curvature

Jouni Parkkonen and Frédéric Paulin

Geometry & Topology 14 (2010) 277–392

DOI: 10.2140/gt.2010.14.277

Bibliography

1	W Abikoff, B Maskit, Geometric decompositions of Kleinian groups, Amer. J. Math. 99 (1977) 687 MR0480992
2	L V Ahlfors, Möbius transformations and Clifford numbers, from: "Differential geometry and complex analysis" (editors I Chavel, H M Farkas), Springer (1985) 65 MR780036
3	C S Aravinda, E Leuzinger, Bounded geodesics in rank-$1$ locally symmetric spaces, Ergodic Theory Dynam. Systems 15 (1995) 813 MR1356615
4	H Aslaksen, Quaternionic determinants, Math. Intelligencer 18 (1996) 57 MR1412993
5	A F Beardon, The geometry of discrete groups, Graduate Texts in Math. 91, Springer (1983) MR698777
6	A Borel, Linear algebraic groups, from: "Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)", Amer. Math. Soc. (1966) 3 MR0204532
7	A Borel, Reduction theory for arithmetic groups, from: "Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)", Amer. Math. Soc. (1966) 20 MR0204533
8	A Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. $(2)$ 75 (1962) 485 MR0147566
9	M Bourdon, Sur le birapport au bord des $\mathrm{CAT}(-1)$–espaces, Inst. Hautes Études Sci. Publ. Math. 83 (1996) 95 MR1423021
10	B H Bowditch, Geometrical finiteness with variable negative curvature, Duke Math. J. 77 (1995) 229 MR1317633
11	M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) MR1744486
12	P Buser, H Karcher, Gromov's almost flat manifolds, Astérisque 81, Société Mathématique de France (1981) 148 MR619537
13	S Buyalo, V Schroeder, M Walz, Geodesics avoiding open subsets in surfaces of negative curvature, Ergodic Theory Dynam. Systems 20 (2000) 991 MR1779390
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) MR1028322
15	T W Cusick, M E Flahive, The Markoff and Lagrange spectra, Math. Surveys and Monogr. 30, Amer. Math. Soc. (1989) MR1010419
16	S G Dani, Bounded orbits of flows on homogeneous spaces, Comment. Math. Helv. 61 (1986) 636 MR870710
17	J Dieudonné, Les déterminants sur un corps non commutatif, Bull. Soc. Math. France 71 (1943) 27 MR0012273
18	J Elstrodt, F Grunewald, J Mennicke, Groups acting on hyperbolic space. Harmonic analysis and number theory, Springer Monogr. in Math., Springer (1998) MR1483315
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 MR1501085
21	G A Freiman, Diophantine approximations and the geometry of numbers (Markov's problem), Kalinin. Gosudarstv. Univ., Kalinin (1975) 144 MR0485714
22	É Ghys, P de la Harpe, editors, Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math. 83, Birkhäuser (1990) MR1086648
23	W M Goldman, Complex hyperbolic geometry, Oxford Math. Monogr., Oxford Univ. Press (1999) MR1695450
24	M Hall Jr., On the sum and product of continued fractions, Ann. of Math. $(2)$ 48 (1947) 966 MR0022568
25	M Hall Jr., The Markoff spectrum, Acta Arith. 18 (1971) 387 MR0296023
26	U Hamenstädt, A new description of the Bowen–Margulis measure, Ergodic Theory Dynam. Systems 9 (1989) 455 MR1016663
27	Y Hellegouarch, Quaternionic homographies: application to Ford hyperspheres, C. R. Math. Rep. Acad. Sci. Canada 11 (1989) 171 MR1010923
28	S Hersonsky, F Paulin, On the volumes of complex hyperbolic manifolds, Duke Math. J. 84 (1996) 719 MR1408542
29	S Hersonsky, F Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv. 72 (1997) 349 MR1476054
30	S Hersonsky, F Paulin, Diophantine approximation for negatively curved manifolds, Math. Z. 241 (2002) 181 MR1930990
31	S Hersonsky, F Paulin, Diophantine approximation in negatively curved manifolds and in the Heisenberg group, from: "Rigidity in dynamics and geometry (Cambridge, 2000)" (editors M Burger, A Iozzi), Springer (2002) 203 MR1919402
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 MR2060999
33	S Hersonsky, F Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds, to appear in J. Diff. Geom. arXiv:0708.3389
34	R Hill, S L Velani, The Jarník–Besicovitch theorem for geometrically finite Kleinian groups, Proc. London Math. Soc. $(3)$ 77 (1998) 524 MR1643409
35	R P Holzapfel, Arithmetische Kugelquotientenflächen I/II, Seminarberichte 14, Humboldt Univ. Sekt. Math. (1978) MR521083
36	R P Holzapfel, Ball and surface arithmetics, Aspects of Math., E29, Friedr. Vieweg & Sohn (1998) MR1685419
37	S Kamiya, On discrete subgroups of $\mathrm{PU}(1,2;\mathbf{C})$ with Heisenberg translations, J. London Math. Soc. $(2)$ 62 (2000) 827 MR1794288
38	R Kellerhals, Quaternions and some global properties of hyperbolic $5$–manifolds, Canad. J. Math. 55 (2003) 1080 MR2005283
39	D Y Kleinbock, G A Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999) 451 MR1719827
40	D Y Kleinbock, B Weiss, Bounded geodesics in moduli space, Int. Math. Res. Not. 30 (2004) 1551 MR2049831
41	A Korányi, H M Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985) 309 MR788413
42	C Maclachlan, A W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Math. 219, Springer (2003) MR1937957
43	C Maclachlan, P L Waterman, N Wielenberg, Higher-dimensional analogues of the modular and Picard groups, Trans. Amer. Math. Soc. 312 (1989) 739 MR965301
44	K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Math. Monogr., Oxford Univ. Press (1998) MR1638795
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 MR1202417
46	J R Parker, Shimizu's lemma for complex hyperbolic space, Internat. J. Math. 3 (1992) 291 MR1146815
47	J Parkkonen, F Paulin, Unclouding the sky of negatively curved manifolds, Geom. Funct. Anal. 15 (2005) 491 MR2153908
48	J Parkkonen, F Paulin, Sur les rayons de Hall en approximation diophantienne, C. R. Math. Acad. Sci. Paris 344 (2007) 611 MR2334070
49	J Parkkonen, F Paulin, Spiraling spectra of geodesic lines in negatively curved manifolds, Preprint (2008) arXiv:0806.1857
50	J Parkkonen, F Paulin, On strictly convex subsets in negatively curved manifolds, in preparation
51	F Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. $(2)$ 54 (1996) 50 MR1395067
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 MR0066431
53	A L Schmidt, Farey simplices in the space of quaternions, Math. Scand. 24 (1969) 31 MR0253993
54	A L Schmidt, On the approximation of quaternions, Math. Scand. 34 (1974) 184 MR0371823
55	T A Schmidt, M Sheingorn, Riemann surfaces have Hall rays at each cusp, Illinois J. Math. 41 (1997) 378 MR1458179
56	V Schroeder, Bounded geodesics in manifolds of negative curvature, Math. Z. 235 (2000) 817 MR1801585
57	C Series, The modular surface and continued fractions, J. London Math. Soc. $(2)$ 31 (1985) 69 MR810563
58	N J A Sloane, editor, The online encyclopedia of integer sequences
59	B Stratmann, The Hausdorff dimension of bounded geodesics on geometrically finite manifolds, Ergodic Theory Dynam. Systems 17 (1997) 227 MR1440778
60	D Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982) 215 MR688349
61	M F Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer (1980) MR580949
62	R Walter, Some analytical properties of geodesically convex sets, Abh. Math. Sem. Univ. Hamburg 45 (1976) 263 MR0417984
63	T Zink, Über die Anzahl der Spitzen einiger arithmetischer Untergruppen unitärer Gruppen, Math. Nachr. 89 (1979) 315 MR546890