Anticonformal Automorphisms Research Articles

On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces

LetX be a Riemann surface of genusg. The surfaceX is called elliptic-hyperelliptic if it admits a conformal involutionh such that the orbit spaceX/〈h〉 has genus one. The involutionh is then called an elliptic-hyperelliptic involution. Ifg>5 then the involutionh is unique, see [A]. We call symmetry to any anticonformal involution ofX. LetAut±(X) be the group of conformal and anticonformal automorphisms ofX and letσ, τ be two symmetries ofX with fixed points and such that {σ, hσ} and {τ, hτ} are not conjugate inAut±(X). We describe all the possible topological conjugacy classes of {σ, σh, τ, τh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genusg (g even >5), the subspace whose elements are the elliptic-hyperelliptic real algebraic curves is not connected. This fact contrasts with the result in [Se]: the subspace whose elements are the hyperelliptic real algebraic curves is connected.

ABELIAN AUTOMORPHISMS GROUPS OF SCHOTTKY TYPE

We study the problem of lifting an Abelian group H of automorphisms of a closed Riemann surface S (containing anticonformals ones) to a suitable Schottky uniformization of S (that is, when H is of Schottky type). If H+ is the index two subgroup of orientation preserving automorphisms of H and R = S/H+, then H induces an anticonformal automorphism ? : R ? R. If ? has fixed points, then we observe that H is of Schottky type. If ? has no fixed points, then we provide a sufficient condition for H to be of Schottky type. We also give partial answers for the excluded cases.

Asymmetric and pseudo-symmetric hyperelliptic surfaces

In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ2. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric.

ALGEBRAIC CURVES, RIEMANN SURFACES AND KLEIN SURFACES WITH NO NON-TRIVIAL AUTOMORPHISMS OR SYMMETRIES

AbstractThe explicit defining equations of a new family of curves whose members have a trivial automorphism group are given. Each member is defined for characteristic zero and all but a finite number of characteristics greater than zero. This family, in conjunction with a previously appearing family of the author’s, provides explicit examples of algebraic curves which possess only the trivial automorphism for each genus greater than three. The family is then used to construct Riemann surfaces without anticonformal automorphisms and Klein surfaces with no non-trivial automorphisms.AMS 2000 Mathematics subject classification: Primary 14H37; 30F50; 30F99

Embeddable anticonformal automorphisms of Riemann surfaces

Let S be a Riemann surface and f be an automorphism of finite order of S. We call f embeddable if there is a conformal embedding $e : S \to \bf {E}^3$ such that $e \circ f \circ e^{-1}$ is the restriction to e(S) of a rigid motion. In this paper we show that an anticonformal automorphism of finite order is embeddable if and only if it belongs to one of the topological conjugation classes here described. For conformal automorphisms a similar result was known by R.A. Rüedy \[R3].

On the topological type of anticonformal square roots of automorphisms of Riemann surfaces

Let X be a compact Riemann surface and f be a conformal automorphism of X of order n. An anticonformal square root of f is an anticonformal automorphism g of X such that g2=f. If g1 and g2 are two anticonformal square roots of f we study the problem of whether g1 and g2 have the same topological type, i. e., if there exists a homeomorphism h:X→X such that g1=hg2h−1. We use techniques of noneuclidean crystallographic (NEC) groups and the topological classification of orientation reversing maps of finite period on surfaces given in [C1] and [Y].

On anticonformal automorphisms of Riemann surfaces with nonembeddable square

In this paper we present an example of an anticonformal automorphism whose square has prime order and is not embeddable. We prove that every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation-reversing self-homeomorphism. Finally we study whether a conformal involution is the square of an orientation-reversing automorphism.

Automorphism groups of complex doubles of Klein surfaces

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+).

Anticonformal automorphisms of compact Riemann surfaces

We show that an automorphism of prime order of a compact Riemann surface is embeddable if it is the square of an anticonformal automorphism. Also, every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation reversing selfhomeomorphism.

Anticonformal Automorphisms of Compact Riemann Surfaces

We show that an automorphism of prime order of a compact Riemann surface is embeddable if it is the square of an anticonformal automorphism. Also, every embeddable automorphism of odd order of a compact Riemann surface is the square of an orientation reversing selfhomeomorphism.

On the number of anticonformal automorphisms of a Riemann surface

On the number of anticonformal automorphisms of a Riemann surface