Dihedral groups are of schottky type
DOI:
https://doi.org/10.22199/S07160917.1999.0001.00003Abstract
We show that a dihedral group H of conforma! automorphisms of a closed Riemann surface S can be lifted for a suitable Schottky uniformization of S. In particular, this implies the existence of a suitable symplectic homology basis of S for which the symplectic representation of H has a simple form.
References
[1] H. M. Farkas. Unramified coverings of hyperelliptic Riemann surfaces. Complex Analysis I, Lecture Notes in Math., vol. 1275, Springer-Verlag, New York, pp. 113-130, (1987).
[2] H. Farkas and I. Kra. Riemann surfaces. Springer-Verlag Graduate Texts in Mathematics, 71, Berlin, Heidelberg, New York, (1991).
[3] F. Harary. Graph theory. Addison-Wesley Series in Mathematics (1969).
[4] R. A. Hidalgo. On Schottky groups with automorphisms. Ann. Acad. Scie. Fenn. Ser. Al Mathematica 19, pp. 247-258, (1994).
[5] R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automorphisms. Glasgow Math. J. 36, pp. 17-32, (1994).
[6] R. A. Hidalgo. Closed Riemann surfaces with dihedral groups of conformal automorphisms. Revista Proyecciones 15, 47-90, (1996).
[7] S. P. Kerckhoff. The Nielsen realization problem. Ann. Math. 117, 235-265, (1983).
[8] B. Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, Vol. 287, Springer- Verlag, Berlín, Heildelberg, New York, (1988).
[9] B. Maskit. On the classification of Kleinian groups I and II. Acta Math. 135 (1975) and 138 (1977).
[10] D. McCullough, A. Miller and B. Zimmermann. Group actions on handlebodies. Proc. London Math. Soc. 59, 373-416, (1989).
[11] J. F. X. Ries. Subvarieties of moduli space determined by finite group actions acting on surfaces. Transactions A.M.S. 335, 385-406, (1993).
[12] S. A. Wolpert. Geodesic length functions and the Nielscn problem. J. Diff. Geom. 25, 275-296, (1987).
[13] B. Zimmermann. Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen. Comment. Math. Helv. 56, 474-486, (1981).
[14] R. Rodríguez and V. González. On principally polarized abelian varieties induced by prysms and pyramids. To appear in Complex Geometry Seminar. Vol. IV (1995).
[2] H. Farkas and I. Kra. Riemann surfaces. Springer-Verlag Graduate Texts in Mathematics, 71, Berlin, Heidelberg, New York, (1991).
[3] F. Harary. Graph theory. Addison-Wesley Series in Mathematics (1969).
[4] R. A. Hidalgo. On Schottky groups with automorphisms. Ann. Acad. Scie. Fenn. Ser. Al Mathematica 19, pp. 247-258, (1994).
[5] R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automorphisms. Glasgow Math. J. 36, pp. 17-32, (1994).
[6] R. A. Hidalgo. Closed Riemann surfaces with dihedral groups of conformal automorphisms. Revista Proyecciones 15, 47-90, (1996).
[7] S. P. Kerckhoff. The Nielsen realization problem. Ann. Math. 117, 235-265, (1983).
[8] B. Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften, Vol. 287, Springer- Verlag, Berlín, Heildelberg, New York, (1988).
[9] B. Maskit. On the classification of Kleinian groups I and II. Acta Math. 135 (1975) and 138 (1977).
[10] D. McCullough, A. Miller and B. Zimmermann. Group actions on handlebodies. Proc. London Math. Soc. 59, 373-416, (1989).
[11] J. F. X. Ries. Subvarieties of moduli space determined by finite group actions acting on surfaces. Transactions A.M.S. 335, 385-406, (1993).
[12] S. A. Wolpert. Geodesic length functions and the Nielscn problem. J. Diff. Geom. 25, 275-296, (1987).
[13] B. Zimmermann. Über Homöomorphismen n-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen. Comment. Math. Helv. 56, 474-486, (1981).
[14] R. Rodríguez and V. González. On principally polarized abelian varieties induced by prysms and pyramids. To appear in Complex Geometry Seminar. Vol. IV (1995).
Published
2018-04-04
How to Cite
[1]
R. A. Hidalgo, “Dihedral groups are of schottky type”, Proyecciones (Antofagasta, On line), vol. 18, no. 1, pp. 23-48, Apr. 2018.
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