Bounds for the order of supersoluble automorphism groups of Riemann surfaces

Author:
Reza Zomorrodian

Journal:
Proc. Amer. Math. Soc. **108** (1990), 587-600

MSC:
Primary 20H10; Secondary 30F20

MathSciNet review:
994795

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Abstract: The maximal automorphism groups of compact Riemann surfaces for a class of groups positioned between nilpotent and soluble groups is investigated. It is proved that if is any finite supersoluble group acting as the automorphism group of some compact Riemann surface of genus , then:

(i) If then and equality occurs when is the supersoluble group that is the semidirect product of the dihedral group of order 8 and the cyclic group of order 3. This exceptional case occurs when the Fuchsian group has the signature (0;2,4,6), and can cover only this finite supersoluble group of order 24.

(ii) If then , and if then must be a power of 3. Conversely if , then there is at least one surface of genus with an automorphism group of order which must be supersoluble since its order is of the form . This bound corresponds to a specific Fuchsian group given by the signature (0;2,3,18). The terms in the chief series of each of these Fuchsian groups to the point where a torsion-free subgroup is reached are computed.

**[1]**W. Burnside,*Theory of groups of finite order*, Dover Publications, Inc., New York, 1955. 2d ed. MR**0069818****[2]**H. S. M. Coxeter and W. O. J. Moser,*Generators and relations for discrete groups*, 3rd ed., Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14. MR**0349820****[3]**K. W. Gruenberg,*Residual properties of infinite soluble groups*, Proc. London Math. Soc. (3)**7**(1957), 29–62. MR**0087652****[4]**Ravi S. Kulkarni,*Symmetries of surfaces*, Topology**26**(1987), no. 2, 195–203. MR**895571**, 10.1016/0040-9383(87)90059-0**[5]**Ravi S. Kulkarni,*Normal subgroups of Fuchsian groups*, Quart. J. Math. Oxford Ser. (2)**36**(1985), no. 143, 325–344. MR**800765**, 10.1093/qmath/36.3.325**[6]**A. M. Macbeath,*On a theorem of Hurwitz*, Proc. Glasgow Math. Assoc.**5**(1961), 90–96 (1961). MR**0146724****[7]**A. M. Macbeath,*Residual nilpotency of Fuchsian groups*, Illinois J. Math.**28**(1984), no. 2, 299–311. MR**740620****[8]**Chih-han Sah,*Groups related to compact Riemann surfaces*, Acta Math.**123**(1969), 13–42. MR**0251216****[9]**Carl Ludwig Siegel,*Some remarks on discontinuous groups*, Ann. of Math. (2)**46**(1945), 708–718. MR**0014088****[10]**David Singerman,*Subgroups of Fuschian groups and finite permutation groups*, Bull. London Math. Soc.**2**(1970), 319–323. MR**0281805****[11]**Henry G. Bray, W. E. Deskins, David Johnson, John F. Humphreys, B. M. Puttaswamaiah, Paul Venzke, and Gary L. Walls,*Between nilpotent and solvable*, Polygonal Publ. House, Washington, N. J., 1982. Edited and with a preface by Michael Weinstein. MR**655785****[12]**Reza Zomorrodian,*Nilpotent automorphism groups of Riemann surfaces*, Trans. Amer. Math. Soc.**288**(1985), no. 1, 241–255. MR**773059**, 10.1090/S0002-9947-1985-0773059-6**[13]**Reza Zomorrodian,*Classification of 𝑝-groups of automorphisms of Riemann surfaces and their lower central series*, Glasgow Math. J.**29**(1987), no. 2, 237–244. MR**901670**, 10.1017/S0017089500006881

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DOI:
https://doi.org/10.1090/S0002-9939-1990-0994795-X

Keywords:
Fuchsian groups,
supersoluble automorphism groups,
nilpotent automorphism groups,
maximal automorphism groups,
compact Riemann surfaces,
action of groups,
generators,
relations,
signatures,
bounds,
chief series

Article copyright:
© Copyright 1990
American Mathematical Society