On automorphisms of graphs and Riemann surfaces acting with fixed points

2019 ◽  
Vol 9 (4) ◽  
pp. 2021-2031
Author(s):  
G. Gromadzki ◽  
A. D. Mednykh ◽  
I. A. Mednykh
Keyword(s):  
Fixed Points ◽  
Riemann Surfaces ◽  
Automorphisms Of Graphs

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.

scholarly journals Fixed Points on Asymmetric Riemann Surfaces

2020 ◽  
Vol 17 (4) ◽  
Author(s):  
Ewa Kozłowska-Walania ◽  
Ewa Tyszkowska
Keyword(s):  
Fixed Points ◽  
Riemann Surfaces

Symmetric Riemann surfaces with no points fixed by orientation preserving automorphisms

2020 ◽  
Vol 126 (3) ◽  
pp. 479-492
Author(s):  
Ewa Kozłowska-Walania
Keyword(s):  
Finite Group ◽  
Fixed Points ◽  
Upper Bound ◽  
Riemann Surfaces ◽  
Dihedral Groups ◽  
Minimal Genus

We study the symmetric Riemann surfaces for which the group of orientation preserving automorphisms acts without fixed points. We show that any finite group can give rise to such an action, determine the maximal number of non-conjugate symmetries for such surfaces and find a sharp upper bound on maximal total number of ovals for a set of $k$ symmetries with ovals. We also solve the minimal genus problem for dihedral groups acting on the surfaces described above, for odd genera.

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.

scholarly journals ON FIXED POINTS OF DOUBLY SYMMETRIC RIEMANN SURFACES

2008 ◽  
Vol 50 (3) ◽  
pp. 371-378 ◽  
Author(s):  
GRZEGORZ GROMADZKI ◽  
EWA KOZŁOWSKA-WALANIA
Keyword(s):  
Fixed Points ◽  
Upper Bound ◽  
Riemann Surfaces ◽  
Upper Bounds

AbstractIn this paper, we study ovals of symmetries and the fixed points of their products on Riemann surfaces of genus g ≥ 2. We show how the number of these points affects the total number of ovals of symmetries. We give a generalisation of Bujalance, Costa and Singerman's theorems in which we show upper bounds for the total number of ovals of two symmetries in terms of g, the order n and the number m of the fixed points of their product, and we show their attainments for n holding some divisibility conditions. Finally, we give an upper bound for m in terms of n and g, and we study conditions under which it has given parity.

scholarly journals Transitive permutation groups with trivial four point stabilizers

2015 ◽  
Vol 18 (5) ◽  
Author(s):  
Kay Magaard ◽  
Rebecca Waldecker
Keyword(s):  
Fixed Points ◽  
Riemann Surfaces ◽  
Permutation Groups ◽  
Weierstrass Points ◽  
Detailed Classification ◽  
Point Stabilizer ◽  
The Relationship ◽  
Automorphisms Of Riemann Surfaces

AbstractIn this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular, we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstraß points and is a continuation of the authors' earlier work.

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