Symmetries of Riemann surfaces with large automorphism group

1974 ◽  
Vol 210 (1) ◽  
pp. 17-32 ◽  
Author(s):  
David Singerman
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Large Automorphism Group

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 Contact metric manifolds with large automorphism group and (κ, µ)-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 294-302 ◽  
Author(s):  
Antonio Lotta
Keyword(s):  
Automorphism Group ◽  
Symmetric Operator ◽  
Homogeneous Spaces ◽  
Point Of View ◽  
Maximum Dimension ◽  
Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Large Automorphism Group ◽  
Dimension 2 ◽  
Simply Connected

AbstractWe discuss the classifiation of simply connected, complete (κ, µ)-spaces from the point of view of homogeneous spaces. In particular, we exhibit new models of (κ, µ)-spaces having Boeckx invariant -1. Finally, we prove that the number ${{(n + 1)(n + 2)} \over 2}$ is the maximum dimension of the automorphism group of a contact metric manifold of dimension 2n +1, n ≥ 2, whose symmetric operator h has rank at least 3 at some point; if this dimension is attained, and the dimension of the manifold is not 7, it must be a (κ, µ)-space. The same conclusion holds also in dimension 7 provided the almost CR structure of the contact metric manifold under consideration is integrable.

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.

scholarly journals A family of conformally asymmetric Riemann surfaces

1997 ◽  
Vol 39 (2) ◽  
pp. 221-225 ◽  
Author(s):  
Brent Everitt
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Coset Diagrams ◽  
Free Subgroups ◽  
Torsion Free

AbstractWe give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

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.

scholarly journals On Riemann surfaces with maximal automorphism groups

1967 ◽  
Vol 8 (2) ◽  
pp. 102-112 ◽  
Author(s):  
Joseph Lehner ◽  
Morris Newman
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Automorphism Groups ◽  
Universal Covering ◽  
Maximum Order ◽  
Nonabelian Group ◽  
Closed Riemann Surface ◽  
Finite Factor ◽  
Upper Half Plane ◽  
Group 2

Let S be a closed Riemann surface of genusg > 1,so that Ŝ, the universal covering surface of S, is hyperbolic. We can then uniformize S by a discrete, nonabelian group Γ1 of Möbius transformations of the upper half-plane ℋ. It follows that N1 = NΩ(Γ1) is discrete; here N1is the normalizer of Γ in Ω, the group of (conformal) automorphisms of ℋ. An automorphism of S can be lifted to a coset of Nl/Γl. Hence C(S), the group of automorphisms of S, is isomorphic to Nl/Γ1. The order of C = C(S) equals the index of Γ1 in N1, which in turn equals ⃒Γ1⃒ / ⃒Nl⃒, where ⃒Nl⃒ is the hyperbolic area of a fundamental region of Nl. Since Γ1 uniformizes a surface, we have ⃒Γ1⃒ = 4π(g – 1), while, by Siegel's results [7], ⃒N1 ⃒ ≧ π/21 and N1 can only be the triangle group (2, 3, 7). Hence in all cases the order of C(S) is at most 84(g–1), an old result of Hurwitz [1]. The surfaces that permit a maximal automorphism group (= automorphism group of maximum order) can therefore be obtained by studying the finite factor groups of (2, 3, 7). Such a treatment, purely algebraic in nature, has been promised by Macbeath [5].

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 Real division algebras with large automorphism group

2004 ◽  
Vol 282 (2) ◽  
pp. 758-796 ◽  
Author(s):  
Dragomir Ž. Đoković ◽  
Kaiming Zhao
Keyword(s):  
Automorphism Group ◽  
Division Algebras ◽  
Large Automorphism Group
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