Fixed Points of Automorphisms of Compact Riemann Surfaces

1970 ◽  
Vol 22 (5) ◽  
pp. 922-932 ◽  
Author(s):  
M. J. Moore
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Riemann Surfaces ◽  
Homology Group ◽  
Group Of Automorphisms ◽  
Quartic Curve ◽  
Lefschetz Fixed Point Formula ◽  
Compact Riemann Surfaces ◽  
Klein's Quartic Curve ◽  
Fundamental Paper

In his fundamental paper [3], Hurwitz showed that the order of a group of biholomorphic transformations of a compact Riemann surface S into itself is bounded above by 84(g – 1) when S has genus g ≧ 2. This bound on the group of automorphisms (as we shall call the biholomorphic self-transformations) is attained for Klein's quartic curve of genus 3 [4] and, from this, Macbeath [7] deduced that the Hurwitz bound is attained for infinitely many values of g.After genus 3, the next smallest genus for which the bound is attained is the case g = 7. The equations of such a curve of genus 7 were determined by Macbeath [8] who also gave the equations of the transformations. The equations of these transformations were found by using the Lefschetz fixed point formula. If the number of fixed points of each element of a group of automorphisms is known, then the Lefschetz fixed point formula may be applied to deduce the character of the representation given by the group acting on the first homology group of the surface.

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|

On fixed points of involutions of compact Riemann surfaces

2009 ◽  
Vol 105 (1) ◽  
pp. 16 ◽  
Author(s):  
E. Bujalance ◽  
G. Gromadzki ◽  
E. Tyszkowska
Keyword(s):  
Fixed Points ◽  
Riemann Surfaces ◽  
Topological Type ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Compact Riemann Surfaces

We find a bound for the total number of fixed points of $k$ commuting involutions of compact Riemann surfaces and we study its attainment. We also find a bound for such number for a pair of non-commuting involutions in terms of the order of their product and the genus of the surface. Finally, we study its attainment, topological type of the action of such pair and the nature of the locus of corresponding surfaces in Teichmüller space.

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3157-3172
Author(s):  
Mujahid Abbas ◽  
Bahru Leyew ◽  
Safeer Khan
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Metric Space ◽  
Delay Differential Equations ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Periodic Solutions ◽  
Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Coincidence and fixed points for hybrid tangential maps

2010 ◽  
Vol 17 (2) ◽  
pp. 273-285
Author(s):  
Tayyab Kamran ◽  
Quanita Kiran
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Common Property ◽  
Fixed Point Theorems ◽  
Acta Math ◽  
Contractive Condition ◽  
The Common

Abstract In [Int. J. Math. Math. Sci. 2005: 3045–3055] by Liu et al. the common property (E.A) for two pairs of hybrid maps is defined. Recently, O'Regan and Shahzad [Acta Math. Sin. (Engl. Ser.) 23: 1601–1610, 2007] have introduced a very general contractive condition and obtained some fixed point results for hybrid maps. We introduce a new property for pairs of hybrid maps that contains the property (E.A) and obtain some coincidence and fixed point theorems that extend/generalize some results from the above-mentioned papers.

scholarly journals Solution of Some Impulsive Differential Equations via Coupled Fixed Point

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 501
Author(s):  
Ahmed Boudaoui ◽  
Khadidja Mebarki ◽  
Wasfi Shatanawi ◽  
Kamaleldin Abodayeh
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Metric Space ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
System Of Differential Equations ◽  
Coupled Fixed Points

In this article, we employ the notion of coupled fixed points on a complete b-metric space endowed with a graph to give sufficient conditions to guarantee a solution of system of differential equations with impulse effects. We derive recisely some new coupled fixed point theorems under some conditions and then apply our results to achieve our goal.

A Hodge theoretic projective structure on compact Riemann surfaces

2021 ◽  
Vol 149 ◽  
pp. 1-27
Author(s):  
Indranil Biswas ◽  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Gian Pietro Pirola
Keyword(s):  
Riemann Surfaces ◽  
Projective Structure ◽  
Compact Riemann Surfaces

scholarly journals Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

2021 ◽  
Vol 24 (3) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Ulrike Bücking
Keyword(s):  
Error Estimate ◽  
Branch Point ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Convergence Result ◽  
Edge Length ◽  
Ramified Covering ◽  
Compact Riemann Surfaces ◽  
Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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