|
Compact non-orientable hyperbolic surfaces with an extremal metric disc
Author(s):
Ernesto
Girondo;
Gou
Nakamura
Journal:
Conform. Geom. Dyn.
11
(2007),
29-43.
MSC (2000):
Primary 30F50;
Secondary 30F40
Posted:
March 8, 2007
MathSciNet review:
2295996
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
The size of a metric disc embedded in a compact non-orientable hyperbolic surface is bounded by some constant depending only on the genus  . We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case  , we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.
References:
-
- [Ba]
- C. Bavard, Disques extrémaux et surfaces modulaires, Ann. de la Fac. des Sciences de Toulouse V (1996), no. 2, 191-202. MR 1413853 (97i:30059)
- [Bu-Ga-Gr]
- E. Bujalance, J.M. Gamboa, and G. Gromadzki, The full automorphism groups of hyperelliptic Riemann surfaces, Manuscripta Math. 79 (1993), 267-282. MR 1223022 (94f:20093)
- [Es-Iz]
- J.L. Estévez and M. Izquierdo, Non-normal pairs of non-Euclidean crystallographic groups, Bull. London Math. Soc. 38 (2006), no. 1, 113-123. MR 2201610 (2006k:20103)
- [Gi-Go1]
- E. Girondo and G. González-Diez, On extremal discs inside compact hyperbolic surfaces, C. R. Acad. Sci. Paris, t. 329, Série I (1999), 57-60. MR 1703263 (2000f:53036)
- [Gi-Go2]
- E. Girondo and G. González-Diez, On extremal Riemann surfaces and their uniformizing Fuchsian groups, Glasgow Math. J. 44 (2002), 149-157. MR 1892291 (2002m:30053)
- [Gi-Go3]
- E. Girondo and G. González-Diez, Genus two extremal surfaces: extremal discs, isometries and Weierstrass points, Israel J. Math. 132 (2002), 221-238. MR 1952622 (2003k:57020)
- [Jo-Na]
- T. Jørgensen and M. Näätänen, Surfaces of genus 2: generic fundamental polygons, Quart. J. Math. Oxford (2), 33 (1982), 451-461. MR 679814 (84c:51029)
- [Mac]
- A.M. Macbeath, Generic Dirichlet polygons and the modular group, Glasgow Math. J., 27 (1985), 129-141. MR 819834 (87b:20062)
- [Mar]
- G. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag 1991. MR 1090825 (92h:22021)
- [Na1]
- G. Nakamura, The number of extremal disks embedded in compact Riemann surfaces of genus two, Sci. Math. Japon 56 (2002), no. 3, 481-492. MR 1937911 (2003i:30065)
- [Na2]
- G. Nakamura, Generic fundamental polygons for surfaces of genus three, Kodai Math. J. 27 (2004), no. 1, 88-104. MR 2042793 (2005a:30069)
- [Na3]
- G. Nakamura, Extremal disks and extremal surfaces of genus three, Kodai Math. J. 28 (2005), no. 1, 111-130. MR 2122195 (2005j:30055)
- [Si]
- D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29-38. MR 0322165 (48:529)
- [Ta1]
- K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan, 29 (1977), 91-106. MR 0429744 (55:2754)
- [Ta2]
- K. Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. (1) 24 (1977), 201-212. MR 0463116 (57:3077)
Similar Articles:
Retrieve articles in Conformal Geometry and Dynamics
with MSC
(2000):
30F50,
30F40
Retrieve articles in all Journals with MSC
(2000):
30F50,
30F40
Additional Information:
Ernesto
Girondo
Affiliation:
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Address at time of publication:
Departamento de Matemáticas, IMAFF, CSIC, Madrid, Spain
Email:
ernesto.girondo@uam.es
Gou
Nakamura
Affiliation:
Science Division, Center for General Education, Aichi Institute of Technology, Yakusa-Cho, Toyota 470-0392, Japan
Email:
gou@aitech.ac.jp
DOI:
10.1090/S1088-4173-07-00157-9
PII:
S 1088-4173(07)00157-9
Keywords:
Extremal discs,
Kleinian surfaces
Received by editor(s):
September 5, 2006
Posted:
March 8, 2007
Additional Notes:
The first author was supported in part by the MCyT research project BFM2003-04964.
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
