scholarly journals Automorphisms of compact non-orientable Riemann surfaces

1971 ◽  
Vol 12 (1) ◽  
pp. 50-59 ◽  
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.

On the Monodromy Groups of Riemann Surfaces of Genus ≧1

1981 ◽  
Vol 33 (5) ◽  
pp. 1142-1156
Author(s):  
Kathryn Kuiken
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Abelian Groups ◽  
General Question ◽  
Group Of Automorphisms ◽  
Cyclic Groups ◽  
Monodromy Groups ◽  
Groups Of Automorphisms ◽  
A Finite Group

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.

On Soluble Groups of Automorphism of Riemann Surfaces

1991 ◽  
Vol 34 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Grzegorz Gromadzki
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Positive Integer ◽  
Riemann Surfaces ◽  
Soluble Group ◽  
Compact Riemann Surface ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Derived Length

AbstractLet G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

Complete set of Genera of Compact Riemann Surfaces with reference to the point Group of Sulphur (S8) Molecule

2020 ◽  
Vol 32 (7) ◽  
pp. 88-92
Author(s):  
RAFIQUL ISLAM ◽  
◽  
CHANDRA CHUTIA ◽  
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Point Group ◽  
Finite Point ◽  
Group Of Automorphisms ◽  
Complete Set ◽  
Compact Riemann Surfaces ◽  
Group Of Symmetries

In this paper we consider the group of symmetries of the Sulphur molecule (S8 ) which is a finite point group of order 16 denote by D16 generated by two elements having the presentation { u\upsilon/u2= \upsilon8 = (u\upsilon)2 = 1} and find the complete set of genera (g ≥ 2) of Compact Riemann surfaces on which D16 acts as a group of automorphisms as follows: D16 the group of symmetries of the sulphur (S8) molecule of order 16 acts as an automorphism group of a compact Riemann surfaces of genus g ≥ 2 if and only if there are integers \lambda and \mu such that \lambda \leq 1 and \mu \geq 1 and g=\lambda +8\mu (\geq2) , \mu\geq |\lambda|

scholarly journals A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group

2020 ◽  
Vol 27 (02) ◽  
pp. 247-262
Author(s):  
Eslam Badr
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Riemann Surfaces ◽  
Real Plane ◽  
Plane Model ◽  
Field Of Moduli ◽  
Field Of Definition ◽  
Smooth Plane ◽  
Real Riemann Surfaces ◽  
Even Order

A Riemann surface [Formula: see text] having field of moduli ℝ, but not a field of definition, is called pseudo-real. This means that [Formula: see text] has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d ≥ 4 in [Formula: see text]. We characterize pseudo-real-plane Riemann surfaces [Formula: see text], whose conformal automorphism group Aut+([Formula: see text]) is PGL3(ℂ)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of [Formula: see text]. In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut+([Formula: see text]) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p.

ALTERNATING GROUPS AS AUTOMORPHISM GROUPS OF RIEMANN SURFACES

2006 ◽  
Vol 16 (01) ◽  
pp. 91-98
Author(s):  
J. J. ETAYO GORDEJUELA ◽  
E. MARTÍNEZ
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Upper Bound ◽  
Riemann Surfaces ◽  
Automorphism Groups ◽  
Alternating Groups ◽  
Minimal Genus

In this work we give pairs of generators (x, y) for the alternating groups An, 5 ≤ n ≤ 19, such that they determine the minimal genus of a Riemann surface on which An acts as the automorphism group. Using these results we prove that A15 is the unique of these groups that is an H*-group, i.e., the groups achieving the upper bound of the order of an automorphism group acting on non-orientable unbordered surfaces.

scholarly journals p-Groups of automorphisms of compact non-orientable Riemann surfaces

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
E. Bujalance ◽  
F. J. Cirre ◽  
J. M. Gamboa
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Upper Bounds ◽  
Full Automorphism Group ◽  
Number Of Generators ◽  
Groups Of Automorphisms ◽  
Topological Genus

AbstractWe study p-groups of automorphisms of compact non-orientable Riemann surfaces of topological genus $$g\ge 3$$ g ≥ 3 . We obtain upper bounds of the order of such groups in terms of p,  g and the minimal number of generators of the group. We also determine those values of g for which these bounds are sharp. Furthermore, the same kind of results are obtained when the p-group acts as the full automorphism group of the surface.

Filling words in fundamental groups of Riemann surfaces

2013 ◽  
Vol 50 (1) ◽  
pp. 31-50
Author(s):  
C. Zhang
Keyword(s):  
Riemann Surface ◽  
Fundamental Group ◽  
Riemann Surfaces ◽  
Fundamental Groups ◽  
Closed Geodesics ◽  
New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

scholarly journals SUPERGRAVITY DESCRIPTION OF FIELD THEORIES ON CURVED MANIFOLDS AND A NO GO THEOREM

2001 ◽  
Vol 16 (05) ◽  
pp. 822-855 ◽  
Author(s):  
JUAN MALDACENA ◽  
CARLOS NUÑEZ
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Field Theories ◽  
De Sitter ◽  
Curved Manifolds ◽  
Conformal Field ◽  
Low Energies ◽  
M Theory

In the first part of this paper we find supergravity solutions corresponding to branes on worldvolumes of the form Rd×Σ where Σ is a Riemann surface. These theories arise when we wrap branes on holomorphic Riemann surfaces inside K3 or CY manifolds. In some cases the theory at low energies is a conformal field theory with two less dimensions. We find some non-singular supersymmetric compactifications of M-theory down to AdS5. We also propose a criterion for permissible singularities in supergravity solutions. In the second part of this paper, which can be read independently of the first, we show that there are no non-singular Randall-Sundrum or de-Sitter compactifications for large class of gravity theories.

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