On Cyclic Groups of Automorphisms of Riemann Surfaces

It is well-known [5, 19] that every finite group can appear as a group of automorphisms of an algebraic Riemann surface. Hurwitz [9, 10] showed that the order of such a group can never exceed 84 (g – 1) provided that the genus g is ≧2. In fact, he showed that this bound is the best possible since groups of automorphisms of order 84 (g – 1) are obtainable for some surfaces of genus g. The problems considered by Hurwitz and others can be considered as particular cases of a more general question: Given a finite group G, what is the minimum genus of the surface for which it is a group of automorphisms? This question has been completely answered for cyclic groups by Harvey [7]. Wiman's bound 2(2g + 1), the best possible, materializes as a consequence. A further step was taken by Maclachlan who answered this question for non-cyclic Abelian groups.

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Groups of automorphisms of Riemann surfaces and maps of genus p + 1 where p is prime

Annales Fennici Mathematici ◽

10.5186/aasfm.2021.4649 ◽

2021 ◽

Vol 46 (2) ◽

pp. 839-867

Author(s):

Milagros Izquierdo ◽

Gareth A. Jones ◽

Sebastián Reyes-Carocca

Keyword(s):

Riemann Surfaces ◽

Groups Of Automorphisms ◽

Automorphisms Of Riemann Surfaces

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On symmetries of compact Riemann surfaces with cyclic groups of automorphisms

Journal of Algebra ◽

10.1016/j.jalgebra.2006.03.019 ◽

2006 ◽

Vol 301 (1) ◽

pp. 82-95 ◽

Cited By ~ 2

Author(s):

E. Bujalance ◽

F.J. Cirre ◽

J.M. Gamboa ◽

G. Gromadzki

Keyword(s):

Riemann Surfaces ◽

Cyclic Groups ◽

Groups Of Automorphisms ◽

Compact Riemann Surfaces

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Supersoluble groups of automorphisms of compact Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500007886 ◽

1989 ◽

Vol 31 (3) ◽

pp. 321-327 ◽

Cited By ~ 4

Author(s):

Grzegorz Gromadzki ◽

Colin MacLachlan

Keyword(s):

Finite Groups ◽

Riemann Surfaces ◽

Nilpotent Groups ◽

Metabelian Groups ◽

Supersoluble Groups ◽

Soluble Groups ◽

Cyclic Groups ◽

Groups Of Automorphisms ◽

Compact Riemann Surfaces ◽

Infinite Sequences

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

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Automorphisms of compact non-orientable Riemann surfaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500001142 ◽

1971 ◽

Vol 12 (1) ◽

pp. 50-59 ◽

Cited By ~ 30

Author(s):

D. Singerman

Keyword(s):

Riemann Surface ◽

Automorphism Group ◽

Riemann Surfaces ◽

Group Of Automorphisms ◽

Groups Of Automorphisms ◽

Definition Of ◽

Orientable Surfaces

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

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