scholarly journals On the order of automorphism groups of Klein surfaces

1985 ◽  
Vol 26 (1) ◽  
pp. 75-81 ◽  
Author(s):  
J. J. Etayo Gordejuela
Keyword(s):  
General Result ◽  
Special Interest ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Automorphism Groups ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Group Of Automorphisms

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

scholarly journals Automorphism groups of complex doubles of Klein surfaces

1994 ◽  
Vol 36 (3) ◽  
pp. 313-330 ◽  
Author(s):  
E. Bujalance ◽  
A. F. Costa ◽  
G. Gromadzki ◽  
D. Singerman
Keyword(s):  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Automorphism Groups ◽  
Natural Question ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Maximal Symmetry ◽  
Group Of Automorphisms

In this paper we consider complex doubles of compact Klein surfaces that have large automorphism groups. It is known that a bordered Klein surface of algebraic genus g > 2 has at most 12(g − 1) automorphisms. Surfaces for which this bound is sharp are said to have maximal symmetry. The complex double of such a surface X is a compact Riemann surface X+ of genus g and it is easy to see that if G is the group of automorphisms of X then C2 × G is a group of automorphisms of X+. A natural question is whether X+ can have a group that strictly contains C2 × G. In [8] C. L. May claimed the following interesting result: there is a unique Klein surface X with maximal symmetry for which Aut X+ properly contains C2 × Aut X (where Aut X+ denotes the group of conformal and anticonformal automorphisms of X+).

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 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).

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 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]).

scholarly journals Large automorphism groups of compact Klein surfaces with boundary, I

1977 ◽  
Vol 18 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Coy L. May
Keyword(s):  
Automorphism Groups ◽  
Klein Surface ◽  
Klein Surfaces

LetXbe a Klein surface [1], that is,Xis a surface with boundary მXtogether with a dianalytic structure onX. A homeomorphismf:X→XofXonto itself that is dianalytic will be called anautomorphismofX.

scholarly journals q-hyperelliptic compact nonorientable Klein surfaces without boundary

2002 ◽  
Vol 31 (4) ◽  
pp. 215-227
Author(s):  
J. A. Bujalance ◽  
B. Estrada
Keyword(s):  
Automorphism Group ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Crystallographic Groups ◽  
Nonorientable Surface

LetXbe a nonorientable Klein surface (KS in short), that is a compact nonorientable surface with a dianalytic structure defined on it. A Klein surfaceXis said to beq-hyperellipticif and only if there exists an involutionΦonX(a dianalytic homeomorphism of order two) such that the quotientX/〈Φ〉has algebraic genusq.q-hyperelliptic nonorientable KSs without boundary (nonorientable Riemann surfaces) were characterized by means of non-Euclidean crystallographic groups. In this paper, using that characterization, we determine bounds for the order of the automorphism group of a nonorientableq-hyperelliptic Klein surfaceXsuch thatX/〈Φ〉has no boundary and prove that the bounds are attained. Besides, we obtain the dimension of the Teichmüller space associated to this type of surfaces.

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.

scholarly journals Atomic Latin Squares based on Cyclotomic Orthomorphisms

10.37236/1919 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Ian M. Wanless
Keyword(s):  
Finite Fields ◽  
Complete Graph ◽  
Prime Order ◽  
Automorphism Groups ◽  
Latin Square ◽  
Latin Squares ◽  
Established Method ◽  
Cyclic Groups ◽  
Complete Bipartite Graphs ◽  
First Time

Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect $1$-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect $1$-factorisations of the complete graph $K_{q+1}$ for many prime powers $q$. As a result, existence of such a factorisation is shown for the first time for $q$ in $\big\{$529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 29929, 32041, 38809, 44521, 50653, 51529, 52441, 63001, 72361, 76729, 78125, 79507, 103823, 148877, 161051, 205379, 226981, 300763, 357911, 371293, 493039, 571787$\big\}$. We show that latin squares built by the 'orthomorphism method' have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples.

Sign in / Sign up

Export Citation Format

Share Document