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 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 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 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 The symmetric genus of 2-groups

1999 ◽  
Vol 41 (1) ◽  
pp. 115-124 ◽  
Author(s):  
JAY ZIMMERMAN
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Compact Riemann Surface ◽  
Group Of Automorphisms ◽  
A Finite Group

A finite group G can be represented as a group of automorphisms of a compact Riemann surface, that is, G acts on a Riemann surface. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts (possibly reversing orientation).

On Soluble Groups of Automorphism of Riemann Surfaces

1991 ◽  
Vol 34 (1) ◽  
pp. 67-73 ◽  
Author(s):  
Grzegorz Gromadzki
Keyword(s):  
Riemann Surface ◽  
Automorphism Group ◽  
Positive Integer ◽  
Riemann Surfaces ◽  
Soluble Group ◽  
Compact Riemann Surface ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Derived Length

AbstractLet G be a soluble group of derived length 3. We show in this paper that if G acts as an automorphism group on a compact Riemann surface of genus g ≠ 3,5,6,10 then it has at most 24(g — 1) elements. Moreover, given a positive integer n we show the existence of a Riemann surface of genus g = n4 + 1 that admits such a group of automorphisms of order 24(g — 1), whilst a surface of specified genus can admit such a group of automorphisms of order 48(g — 1), 40(g — 1), 30(g — 1) and 36(g — 1) respectively.

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.

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