scholarly journals Topological types of Klein surfaces with a maximum order automorphism

1988 ◽  
Vol 30 (1) ◽  
pp. 87-96
Author(s):  
J. A. Bujalance
Keyword(s):  
Boundary Component ◽  
Topological Type ◽  
Klein Surface ◽  
Maximum Order ◽  
Klein Surfaces ◽  
Order Automorphism

If X is a Klein surface (KS) with boundary, of algebraic genus p, and Φ is an automorphism of order N, May [8] proved that N ≤ 2p + 2 when X is orientable and p is even, and N ≤ 2p otherwise.He proved also that the unique topological type of an orientable KS having an orientation-preserving automorphism of maximum order is a surface with one boundary component when p is even, with two boundary components when p is odd.

scholarly journals PRIME ORDER AUTOMORPHISMS OF KLEIN SURFACES REPRESENTABLE BY ROTATIONS ON THE EUCLIDEAN SPACE

2012 ◽  
Vol 21 (04) ◽  
pp. 1250040 ◽  
Author(s):  
ANTONIO F. COSTA ◽  
CAM VAN QUACH HONGLER
Keyword(s):  
Euclidean Space ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Topological Type ◽  
Klein Surface ◽  
Klein Surfaces

Let S be a bordered orientable Klein surface and p a prime. Assume that f is an order p automorphism of S. In this work we obtain the conditions on the topological type of (S, f) to be conformally equivalent to (S′, f′) where S′ is a bordered orientable Klein surface embedded in the Euclidean space and f′ is the restriction to S′ of a prime order rotation. Our results can allow the visualization of some Riemann surfaces automorphisms by representing them as restrictions of isometries of 𝕊4 or ℝ4. We illustrate this method with the order seven automorphisms of two famous Riemann surfaces: the Klein quartic and the Wiman surface.

scholarly journals DOUBLES OF KLEIN SURFACES

2012 ◽  
Vol 54 (3) ◽  
pp. 507-515
Author(s):  
ANTONIO F. COSTA ◽  
WENDY HALL ◽  
DAVID SINGERMAN
Keyword(s):  
Klein Bottle ◽  
Orientable Surface ◽  
Historical Note ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Algebraic Functions ◽  
Genus 2

Historical note. A non-orientable surface of genus 2 (meaning 2 cross-caps) is popularly known as the Klein bottle. However, the term Klein surface comes from Felix Klein's book “On Riemann's Theory of Algebraic Functions and their Integrals” (1882) where he introduced such surfaces in the final chapter.

The automorphism group of compact Klein surfaces with one boundary component

1990 ◽  
pp. 138-152
Author(s):  
Emilio Bujalance ◽  
José Javier Etayo ◽  
José Manuel Gamboa ◽  
Grzegorz Gromadzki
Keyword(s):  
Automorphism Group ◽  
Boundary Component ◽  
Klein Surfaces

scholarly journals Complex doubles of bordered Klein surfaces with maximal symmetry

1991 ◽  
Vol 33 (1) ◽  
pp. 61-71 ◽  
Author(s):  
Coy L. May
Keyword(s):  
Automorphism Group ◽  
Special Interest ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Maximal Symmetry

A compact bordered Klein surface X of algebraic genus g ≥ 2 has maximal symmetry [6] if its automorphism group A(X) is of order 12(g — 1), the largest possible. The bordered surfaces with maximal symmetry are clearly of special interest and have been studied in several recent papers ([6] and [9] among others).

THE REAL GENUS OF 2-GROUPS

2007 ◽  
Vol 06 (01) ◽  
pp. 103-118 ◽  
Author(s):  
COY L. MAY
Keyword(s):  
Finite Group ◽  
Klein Surface ◽  
Positive Real ◽  
Klein Surfaces ◽  
The Real ◽  
A Finite Group

Let G be a finite group. The real genus ρ(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. Here we consider 2-groups acting on bordered Klein surfaces. The main focus is determining the real genus of each of the 51 groups of order 32. We also obtain some general results about the partial presentations that 2-groups acting on bordered surfaces must have. In addition, we obtain genus formulas for some families of 2-groups and show that if G is a 2-group with positive real genus, then ρ(G) ≡ 1 mod 4.

scholarly journals On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary

2017 ◽  
Vol 28 (05) ◽  
pp. 1750038 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo ◽  
Ana Maria Porto
Keyword(s):  
Moduli Space ◽  
Boundary Component ◽  
Branch Locus ◽  
Klein Surfaces ◽  
Genus Formula

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

scholarly journals THE HYPERELLIPTIC MAPPING CLASS GROUP OF KLEIN SURFACES

2001 ◽  
Vol 44 (2) ◽  
pp. 351-363 ◽  
Author(s):  
E. Bujalance ◽  
A. F. Costa ◽  
J. M. Gamboa
Keyword(s):  
Moduli Space ◽  
Mapping Class Group ◽  
Class Group ◽  
Algebraic Curves ◽  
Topological Type ◽  
Mapping Class ◽  
Klein Surfaces ◽  
Primary 14H10 ◽  
Generators And Relations ◽  
Real Algebraic Curves

AbstractIn this paper we study the algebraic structure of the hyperelliptic mapping class group of Klein surfaces, which is closely related to the mapping class group of punctured discs. This group plays an important role in the study of the moduli space of hyperelliptic real algebraic curves. Our main result provides a presentation by generators and relations for the hyperelliptic mapping class group of surfaces of prescribed topological type.AMS 2000 Mathematics subject classification: Primary 14H10; 20H10; 30F50

scholarly journals Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism

1995 ◽  
Vol 37 (2) ◽  
pp. 221-232 ◽  
Author(s):  
E. Bujalance ◽  
J. M. Gamboa ◽  
C. Maclachlan
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Compact Riemann Surface ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Abelian Case ◽  
Power Automorphism ◽  
Topological Genus

In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

scholarly journals Extremal length and Dirichlet problem on Klein surfaces

2019 ◽  
Vol 39 (2) ◽  
pp. 281-296
Author(s):  
Monica Roşiu
Keyword(s):  
Dirichlet Problem ◽  
Extremal Length ◽  
Extremal Problems ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Conformally Invariant ◽  
Extremal Distance

The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.

scholarly journals Orientable and non-orientable Klein surfaces with maximal symmetry

1985 ◽  
Vol 26 (1) ◽  
pp. 31-34 ◽  
Author(s):  
David Singerman
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Closed Surface ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Maximal Symmetry ◽  
Group Of Automorphisms ◽  
G 12 ◽  
Image Position ◽  
A Finite Group

Let X be a bordered Klein surface, by which we mean a Klein surface with non-empty boundary. X is characterized topologically by its orientability, the number k of its boundary components and the genus p of the closed surface obtained by filling in all the holes. The algebraic genus g of X is defined by.If g≥2 it is known that if G is a group of automorphisms of X then |G|≤12(g-l) and that the upper bound is attained for infinitely many values of g ([4], [5]). A bordered Klein surface for which this upper bound is attained is said to have maximal symmetry. A group of 12(g-l) automorphisms of a bordered Klein surface of algebraic genus g is called an M*-group and it is known that a finite group G is an M*-group if and only if it is generated by 3 non-trivial elements T1, T2, T3 which obey the relations([4]).

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