A new characterization of Gromov hyperbolicity for negatively curved surfaces

Publicacions Matematiques

Volumn 50, Issue 2, 2006, Pages 249-278

  Rodríguez, José M ^a   Touris E V A ^a  

^a UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

Closed geodesic; Gromov hyperbolicity; Hyperbolic riemann surface

Indexed keywords

EID: 33751255404     PISSN: 02141493     EISSN: None     Source Type: Journal    
DOI: 10.5565/PUBLMAT_50206_01     Document Type: Article

References (35)

1. [AR] V. ÁLVAREZ AND J. M. RODRÍGUEZ, Structure theorems for Riemann and topological surfaces, J. London Math. Soc. (2) 69(1) (2004), 153-168.
2. [ARY] V. ÁLVAREZ, J. M. RODRÍGUEZ AND D. V. YAKUBOVICH, Estimates for nonlinear harmonic "measures" on trees, Michigan Math. J. 49(1) (2001), 47-64.
3. [An] J. W. ANDERSON, "Hyperbolic geometry", Springer Undergraduate Mathematics Series, Springer-Verlag London, Ltd., London, 1999.
4. [BB] Z. M. BALOGH AND S. M. BUCKLEY, Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153(2) (2003), 261-301.
5. [B] A. F. BEARDON, "The geometry of discrete groups", Graduate Texts in Mathematics 91, Springer-Verlag, New York, 1983.
6. [Bo] M. BONK, Quasi-geodesic segments and Gromov hyperbolic spaces, Geom. Dedicata 62(3) (1996), 281-298.
7. [BHK] M. BONK, J. HEINONEN AND P. KOSKELA, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), 99 pp.
8. [Bu] P. BUSER, "Geometry and spectra of compact Riemann surfaces", Progress in Mathematics 106, Birkhäuser Boston, Inc., Boston, MA, 1992.
9. [CFPR] A. CANTÓN, J. L. FERNÁNDEZ, D. PESTANA AND J. M. RODRÍGUEZ, On harmonic functions on trees, Potential Anal. 15(3) (2001), 199-244.
10. [C] I. CHAVEL, "Eigenvalues in Riemannian geometry", Including a chapter by Burton Randol, With an appendix by Jozef Dodziuk, Pure and Applied Mathematics 115, Academic Press, Inc.. Orlando, FL, 1984.
11. [CDP] M. COORNAERT, T. DELZANT AND A. PAPADOPOULOS, "Gé ométrie et théorie des groupes. Les groupes hyperboliques de Gromov", Lecture Notes in Mathematics 1441, Springer-Verlag, Berlin, 1990.
12. [F] W. FENCHEL, "Elementary geometry in hyperbolic space", With an editorial by Heinz Bauer, de Gruyter Studies in Mathematics 11, Walter de Gruyter & Co., Berlin, 1989.
13. [FM] J. L. FERNÁNDEZ AND M. V. MELIÁN, Escaping geodesics of Riemannian surfaces, Acta Math. 187(2) (2001), 213-236.
14. [FR1] J. L. FERNÁNDEZ AND J. M. RODRÍGUEZ, The exponent of convergence of Riemann surfaces. Bass Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 15(1) (1990), 165-183.
15. [FR2] J. L. FERNÁNDEZ AND J. M. RODRÍGUEZ, Area growth and Green's function of Riemann surfaces, Ark. Mat. 30(1) (1992), 83-92.
16. [GH] É. GHYS AND P. DE LA HARPE, EDS., "Sur les groupes hyperboliques d'après Mikhael Gromov", Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988, Progress in Mathematics 83, Birkhäuser Boston, Inc., Boston, MA, 1990.
17. [G1] M. GROMOV, Hyperbolic groups, in: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, pp. 75-263.
18. [G2] M. GROMOV, "Metric structures for Riemannian and non-Riemannian spaces", Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates, Progress in Mathematics 152, Birkhäuser Boston, Inc., Boston, MA, 1999.
19. [HS] I. HOLOPAINEN AND P. M. SOARDI, p-harmonic functions on graphs and manifolds, Manuscripta Math. 94(1) (1997), 95-110.
20. [JS] G. A. JONES AND D. SINGERMAN, "Complex functions. An algebraic and geometric viewpoint", Cambridge University Press, Cambridge, 1987.
21. [K1] M. KANAI, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37(3) (1985), 391-413.
22. [K2] M. KANAI, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38(2) (1986), 227-238.
23. [K3] M. KANAI, Analytic inequalities, and rough isometries between noncompact Riemannian manifolds, in: "Curvature and topology of Riemannian manifolds" (Katata, 1985), Lecture Notes in Math. 1201, Springer, Berlin, 1986, pp. 122-137.
24. [N] P. J. NICHOLLS, "The ergodic theory of discrete groups", London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge, 1989.
25. [P] F. PAULIN, On the critical exponent of a discrete group of hyperbolic isometries, Differential Geom. Appl. 7(3) (1997), 231-236.
26. [PRT1] A. PORTILLA, J. M. RODRÍGUEZ AND E. TOURÍS, Gromov hyperbolicity through decomposition of metrics spaces. II, J. Geom. Anal. 14(1) (2004), 123-149.
27. [PRT2] A. PORTILLA, J. M. RODRÍGUEZ AND E. TOURÍS, The topology of balls and Gromov hyperbolicity of Riemann surfaces, Differential Geom. Appl. 21(3) (2004), 317-335.
28. [R] B. RANDOL, Cylinders in Riemann surfaces, Comment. Math. Helv. 54(1) (1979), 1-5.
29. [Ra] J. G. RATCLIFFE, "Foundations of hyperbolic manifolds", Graduate Texts in Mathematics 149, Springer-Verlag, New York, 1994.
30. [R1] J. M. RODRÍGUEZ, Isoperimetric inequalities and Dirichlet functions of Riemann surfaces, Publ. Mat. 38(1) (1994), 243-253.
31. [R2] J. M. RODRÍGUEZ, Two remarks on Riemann surfaces, Publ. Mat. 38(2) (1994), 463-477.
32. [RT1] J. M. RODRÍGUEZ AND E. TOURÍS, Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hungar. 103(1-2) (2004), 107-138.
33. [RT2] J. M. RODRÍGUEZ AND E. TOURÍS, Gromov hyperbolicity of Riemann surfaces, Acta Math. Sin. (Engl. Ser.) (to appear).
34. [S] H. SHIMIZU, On discontinuous groups operating on the product of the upper half planes, Ann. of Math. (2) 77 (1963), 33-71.
35. [So] P.M. SOARDI, Rough isometries and Dirichlet finite harmonic functions on graphs, Proc. Amer. Math. Soc. 119(4) (1993), 1239-1248.