Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

Mediterranean Journal of Mathematics

Volumn 8, Issue 1, 2011, Pages 49-67

  Hästö, Peter ^a   Portilla, Ana ^b   Rodriguez J M ^c   Tourís, Eva ^c  

^a UNIVERSITY OF OULU   (Finland)

^b St Louis University (Madrid Campus)   (Spain)

^c UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

Gromov hyperbolicity; Quasihyperbolic metric; uniformly separated

Indexed keywords

EID: 79751533492     PISSN: 16605446     EISSN: 16605454     Source Type: Journal    
DOI: 10.1007/s00009-010-0059-7     Document Type: Article

References (34)

1. Ahlfors L., Sario L.: Riemann surfaces. Princeton Univ. Press, Princeton (1960).
2. Alvarez V., Pestana D., Rodríguez J. M.: Isoperimetric inequalities in Riemann surfaces of infinite type. Rev. Mat. Iberoamericana 15, 353-427 (1999).
3. Alvarez V., Portilla A., Rodríguez J. M., Tourís E.: Gromov hyperbolicity of Denjoy domains. Geom. Dedicata 121, 221-245 (2006).
4. Anderson J. W.: Hyperbolic Geometry. Springer, London (1999).
5. Balogh Z. M., Bonk M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75, 504-533 (2000).
6. Balogh Z. M., Buckley S. M.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261-301 (2003).
7. Beardon A. F., Pommerenke Ch.: The Poincaré metric of a plane domain. J. London Math. Soc. 18, 475-483 (1978).
8. Benoist Y.: Convexes hyperboliques et fonctions quasisymtriques. Publ. Math. Inst. Hautes Études Sci. 97, 181-237 (2003).
9. M. Bonk, J. Heinonen, P. Koskela, Uniformizing Gromov hyperbolic spaces. Astérisque 270 (2001).
10. M. Bonk, O. Schramm, Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10 (2000), no. 2, 266-306.
11. T. Foertsch, V. Schroeder, A product construction for hyperbolic metric spaces. Illinois J. Math. 49 (2005), no. 3, 793-810.
12. E. Ghys, P. de la Harpe, Sur les Groupes Hyperboliques d'apres Mikhael Gromov. Progress in Mathematics, Volume 83. Birkhäuser. 1990.
13. J. Gonzalo, A. Portilla, J. M. Rodriguez, E. Touris, Uniformly separated sets, Gromov hyperbolicity and the topology of balls. Preprint (2008).
14. M. Gromov, Hyperbolic groups, Essays in group theory, 75-263, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987.
15. Hästö P. A.: Gromov hyperbolicity of the jG and j̃G metrics. Proc. Amer. Math. Soc. 134, 1137-1142 (2006).
16. P. A. Hästö, H. Lindén, A. Portilla, J. M. Rodríguez, E. Tourís, Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics. Preprint (2008).
17. P. A. Hästö, H. Lindén, A. Portilla, J. M. Rodríguez, E. Tourís, Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains. Preprint (2008).
18. P. A. Hästö, H. Lindén, A. Portilla, J. M. Rodríguez, E. Tourís, Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics. To appear in Complex Var. Elliptic Equ.
19. Karlsson A., Noskov G. A.: The Hilbert metric and Gromov hyperbolicity. Enseign. Math. 48, 73-89 (2002).
20. H. Lindén, Gromov hyperbolicity of certain conformal invariant metrics. Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 279-288.
21. A. Naor, Y. Peres, O. Schramm, S. Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces. Duke Math. J. 134 (2006), no. 1, 165-197.
22. Pommerenke Ch.: Uniformly perfect sets and the Poincaré metric. Arch. Math. 32, 192-199 (1979).
23. Portilla A., Rodríguez J. M., Tourís E.: Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal. 14, 123-149 (2004).
24. Portilla A., Rodríguez J. M., Tourís E.: The topology of balls and Gromov hyperbolicity of Riemann surfaces. Diff. Geom. Appl. 21, 317-335 (2004).
25. Portilla A., Rodríguez J. M., Tourís E.: The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces. Proc. Edinburgh Math. Soc. 49, 399-425 (2006).
26. Portilla A., Rodríguez J. M., Tourís E.: Stability of Gromov hyperbolicity. J. Advan. Math. Studies 2, 1-20 (2009).
27. A. Portilla, J. M. Rodríguez, E. Tourís, A real variable characterization of Gromov hyperbolicity of flute surfaces. Preprint (2008).
28. Portilla A., Tourís E.: A characterization of Gromov hyperbolicity of surfaces with variable negative curvature. Publ. Mat. 53, 83-110 (2009).
29. Randol B.: Cylinders in Riemann surfaces. Comment. Math. Helv. 54, 1-5 (1979).
30. Rodríguez J. M., Tourís E.: Gromov hyperbolicity through decomposition of metric spaces. Acta Math. Hung. 103, 53-84 (2004).
31. Rodríguez J. M., Tourís E.: A new characterization of Gromov hyperbolicity for Riemann surfaces. Publ. Mat. 50, 249-278 (2006).
32. Rodríguez J. M., Tourís E.: Gromov hyperbolicity of Riemann surfaces. Acta Math. Sinica 23, 209-228 (2007).
33. Shimizu H.: On discontinuous groups operating on the product of upper halfplanes. Ann. of Math. 77, 33-71 (1963).
34. S. Wenger, Gromov hyperbolic spaces and the sharp isoperimetric constant. Invent. Math. 171 (2008), no. 1, 227-255.