ON CONJUGACY OF p-GONAL AUTOMORPHISMS
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In 1995 it was proved by Gonz$\acute{a}$lez-Diez that the cyclic group generated by a p-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits to obtain that the cyclic group generated by a conformal automorphism of order p of a handlebody with a Kleinian structure and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.
Abstract
In 1995 it was proved by Gonz$\acute{a}$lez-Diez that the cyclic group generated by a p-gonal automorphism of a closed Riemann surface of genus at least two is unique up to conjugation in the full group of conformal automorphisms. Later, in 2008, Gromadzki provided a different and shorter proof of the same fact using the Castelnuovo-Severi theorem. In this paper we provide another proof which is shorter and is just a simple use of Sylow's theorem together with the Castelnuovo-Severi theorem. This method permits to obtain that the cyclic group generated by a conformal automorphism of order p of a handlebody with a Kleinian structure and quotient the three-ball is unique up to conjugation in the full group of conformal automorphisms.
참고문헌 (8)
- G. Castelnuovo, "Sulle serie algebriche si gruppi di punti appartenenti ad una curve al- gebraica", Rend. Real Accad. Lincei (5), 15, pp.509~
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- G. Gonzalez-Diez, "On prime Galois coverings of the Riemann sphere", Ann. Math. Pura Appl. (4), 168, pp.1~15 (1995)
- G. Gromadzki, "On conjugacy of p-gonal automorphisms of Riemann surfaces", Rev. Mat. Complut, 21(1), pp.83~87 (2008)
- R. A. Hidalgo, "Automorphism groups of Schottky type", Ann. Acad. Sci. Fenn. Math, 30(1), pp.183~204 (2005)
- B. Maskit, "Kleinian Groups, GMW" (1987), Springer-Verlag
- D. Singerman, "Finitely maximal Fuchsian groups", J. London Math. Soc. (2), 6, pp.29~38 (1972)
- A. Wootton, "The full automorphism group of a cyclic p-gonal surface", J. Algebra, 312(1), pp.377~396 (2007)