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THE ROLE OF FUNNELS AND PUNCTURES IN THE GROMOV HYPERBOLICITY OF RIEMANN SURFACES

Published online by Cambridge University Press:  30 May 2006

Ana Portilla ,
José M. Rodríguez  and
Eva Tourís
Ana Portilla
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain (apferrei@math.uc3m.es; jomaro@math.uc3m.es; etouris@math.uc3m.es)
José M. Rodríguez
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain (apferrei@math.uc3m.es; jomaro@math.uc3m.es; etouris@math.uc3m.es)
Eva Tourís
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911 Leganés, Madrid, Spain (apferrei@math.uc3m.es; jomaro@math.uc3m.es; etouris@math.uc3m.es)
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Abstract

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We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface $S^*$ obtained by deleting a closed set from one original surface $S$. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

Keywords

Primary 30F2030F45hyperbolicityRiemann surfacefunnelpuncture
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 2 , June 2006 , pp. 399 - 425
Copyright
Copyright © Edinburgh Mathematical Society 2006