Compact non-orientable hyperbolic surfaces with an extremal metric disc

Girondo, Ernesto; Nakamura, Gou

doi:10.1090/S1088-4173-07-00157-9

Compact non-orientable hyperbolic surfaces with an extremal metric disc
HTML articles powered by AMS MathViewer

by Ernesto Girondo and Gou Nakamura

Conform. Geom. Dyn. 11 (2007), 29-43

DOI: https://doi.org/10.1090/S1088-4173-07-00157-9

Published electronically: March 8, 2007

PDF | Request permission

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 $g \ge 3$. We show that a surface of genus greater than six contains at most one metric disc of the largest radius. For the case $g=3$, we carry out an exhaustive study of all the extremal surfaces, finding the location of every extremal disc inside them.

References

Christophe Bavard, Disques extrémaux et surfaces modulaires, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 2, 191–202 (French, with English and French summaries). MR 1413853, DOI 10.5802/afst.827
E. Bujalance, J. M. Gamboa, and G. Gromadzki, The full automorphism groups of hyperelliptic Riemann surfaces, Manuscripta Math. 79 (1993), no. 3-4, 267–282. MR 1223022, DOI 10.1007/BF02568345
José Luis Estévez and Milagros Izquierdo, Non-normal pairs of non-Euclidean crystallographic groups, Bull. London Math. Soc. 38 (2006), no. 1, 113–123. MR 2201610, DOI 10.1112/S0024609305017984
Ernesto Girondo and Gabino González-Diez, On extremal discs inside compact hyperbolic surfaces, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 1, 57–60 (English, with English and French summaries). MR 1703263, DOI 10.1016/S0764-4442(99)80461-3
Ernesto Girondo and Gabino González-Diez, On extremal Riemann surfaces and their uniformizing Fuchsian groups, Glasg. Math. J. 44 (2002), no. 1, 149–157. MR 1892291, DOI 10.1017/S0017089502010108
Ernesto Girondo and Gabino González-Diez, Genus two extremal surfaces: extremal discs, isometries and Weierstrass points, Israel J. Math. 132 (2002), 221–238. MR 1952622, DOI 10.1007/BF02784513
Troels Jørgensen and Marjatta Näätänen, Surfaces of genus $2$: generic fundamental polygons, Quart. J. Math. Oxford Ser. (2) 33 (1982), no. 132, 451–461. MR 679814, DOI 10.1093/qmath/33.4.451
A. M. Macbeath, Generic Dirichlet polygons and the modular group, Glasgow Math. J. 27 (1985), 129–141. MR 819834, DOI 10.1017/S0017089500006133
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
Gou Nakamura, The number of extremal disks embedded in compact Riemann surfaces of genus two, Sci. Math. Jpn. 56 (2002), no. 3, 481–492. MR 1937911
Gou Nakamura, Generic fundamental polygons for surfaces of genus three, Kodai Math. J. 27 (2004), no. 1, 88–104. MR 2042793, DOI 10.2996/kmj/1085143791
Gou Nakamura, Extremal disks and extremal surfaces of genus three, Kodai Math. J. 28 (2005), no. 1, 111–130. MR 2122195, DOI 10.2996/kmj/1111588041
David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. MR 322165, DOI 10.1112/jlms/s2-6.1.29
Kisao Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), no. 1, 91–106. MR 429744, DOI 10.2969/jmsj/02910091
Kisao Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201–212. MR 463116