BOUNDS FOR THE ORDER OF SUPERSOLUBLE AUTOMORPHISM GROUPS OF RIEMANN SURFACES (original) (raw)

ON A THEOREM OF SUPERSOLUBLE AUTOMORPHISM GROUPS

Proceedings of American Mathematical Society, 2002

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc.

NILPOTENT AUTOMORPHISM GROUPS OF RIEMANN SURFACES

Transactions of AMS, 1985

The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automor-phisms of a surface of genus g > 2 cannot exceed 16(g-1). Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0; 2,4,8).

Classification of p-groups of automorphisms of Riemann surfaces and their lower central series

Glasgow Mathematical Journal, 1987

In a previous paper [7], I have made a study of the ”nilpotent” analogue of Hurwitz theorem [4] by considering a particular family of signatures called ”nilpotent admissible” [5]. We saw however, that if μN(g) represents the order of the largest nilpotent group of automorphisms of a surface of genus g < 2, then μN(g) < 16(g − 1) and this upper bound occurs when the covering group is a triangle group having the signature (0; 2,4,8) which is in its own 2-local formThe restriction to the nilpotent groups enabled me to obtain much more precise information than was available in the general case. Moreover, all nilpotent groups attaining this maximum order turned out to be ”2-groups”. Since every finite nilpotent group is the direct product of its Sylow subgroups and the groups of automorphisms are factor groups of the Fuchsian groups, it is natural for us to study the Fuchsian groups havin p-local signatures to obtain more precise information about the finite p-groups, and hence abo...

Nilpotent groups of automorphisms of families of Riemann surfaces

2020

In this article we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d ≥ 1. We provide conditions under which these bounds are sharp and, in addition, for the one-dimensional case we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms.

On compact Riemann surfaces with dihedral groups of automorphisms

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We study compact Riemann surfaces of genus g 2 having a dihedral group of automorphisms. We find necessary and sufficient conditions on the signature of a Fuchsian group for it to admit a surface kernel epimorphism onto the dihedral group D N. The question of extendability of the action of D N is considered. We also give an explicit parametrization of the moduli space of Riemann surfaces with maximal dihedral symmetry, showing that it is a one-dimensional complex manifold. Defining equations of all such surfaces and the formulae of their automorphisms are calculated. The locus of this moduli space consisting of those surfaces admitting some real structure is determined.

AUTOMORPHISM GROUPS OF RIEMANN SURFACES OF GENUS p+1, WHERE p IS PRIME

Glasgow Mathematical Journal, 2005

We show that if S is a compact Riemann surface of genus g = p + 1, where p is prime, with a group of automorphisms G such that |G| ≥ λ(g − 1) for some real number λ > 6, then for all sufficiently large p (depending on λ), S and G lie in one of six infinite sequences of examples. In particular, if λ = 8 then this holds for all p ≥ 17.

Riemann surface with cyclic automorphisms group

Proyecciones (Antofagasta), 1997

In t his paper. we present tllC' uniformization of y 2 = .rP-l, with p > 5 aurl prime. i. e .. the only hyperelliptic Riemann surface of gt'nus (/-7. \\"hich admit Z j2pZ as automorphism group. This 1111ifonnization is fouud by using a fuc:hsian group which rcflects the actiou of Z/2pZ aud is coustructed starting of a triangle group of !YJW (0:¡>.p.p). I\loreover. we describe completely the action of the automorphism group in hmnology. so that we can describe the invariant subvariety for Z /2pZ in A 9 (principally polarized abelian varieties of dimension y). which is detPrmiued bv the real Abe! aplication from M 9 in A 9 .

Symmetries of Riemann surfaces with large automorphism group

Mathematische Annalen, 1974

A Riemann surface is symmetric if it admits an anti-conformal involution. The basic question which we discuss in this paper is whether compact Riemann surfaces of genus g > t which admit large groups of automorphisms are symmetric. As is weU-known, the automorphism group of a compact Riemann surface of genus g > 1 is finite and bounded above by 84(g-1). Macbeath ([t21 13]) has found infinitely many g for which this bound is attained. We show that all the surfaces found by Macbeath's methods are indeed symmetric. However, we do exhibit an example of a non-symmetric Riemann surface of genus g = ! 7 which does admit 84(g-1) automorphisms. We also study Riemann surfaces admitting automorphisms of large order. The order of an automorphism of a Riemann surface of genus g is bounded above by 4g + 2 and this bound is attained for every g [8]. We show that all Riemann surfaces admitting automorphisms of order greater that 2g + 2 are symmetric. There is a close link between our work and the theory of irreflexible regular maps on surfaces. (See § 8 for definitions.) There is a connection between the groups of regular maps and large groups of automorphisms of compact Riemann surfaces. Indeed, every group of automorphisms ofa Riemann surface of genus g of order greater than 24(g-1) is also the group of some regular map and conversely, every group of a regular map can be thought of as the group of automorphisms of a Riemann surface. The irreflexible regular maps turn out to be rather exceptional. (In fact, it was suggested in early editions of [3] that they did not exist for surfaces of genus O > 1). We show in the above correspondence that large groups of automorphisms of non-symmetric surfaces will give rise to irreflexible regular maps, but that the converse of this fact is not always true. Thus, for example, groups of automorphisms of order greater than 24(g-1) of a compact non-symmetric Riemann surface of genus g are more exceptional than irreflexible regular maps. There is another interpretation of symmetric Riemann surfaces which is of interest. Every compact Riemann surface can be obtained as the Riemann surface of an algebraic curve f(z, w) = 0. A Riemann surface