The Symmetry Type of a Riemann Surface (original) (raw)

Pairs of Symmetries of Riemann Surfaces

Annales Academiae Scientiarum Fennicae. Mathematica, 1998

Let Fg be a compact Riemann surface of genus g. A symmetry S of Fg is an anticonformal involution acting on Fg . The fixed-point set of a symmetry is a collection of disjoint simple closed curves, called the mirrors of the symmetry. The number of mirrors |S| of a symmetry of a surface of genus g can be any integer k with 0 ≤ k ≤ g + 1. However, if a Riemann surface Fg admits a symmetry S1 with k mirrors then work of Bujalance and Costa (1) and Natanzon (9) on symmetries with g+1 mirrors suggest that there may possibly be restrictions on the number of mirrors of another symmetry S2 of Fg . In the first three sections of this work we show that the number of such restrictions is few and only occur if one of the symmetries has g+1 or 0 mirrors. The main result of Sections 1-3 is Theorem 1.1 below. In Section 4 we study a finer classification than the number of mirrors, namely the species of a symmetry. The k mirrors of a symmetry S may or may not separate the surface Fg into two non-emp...

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

Israel Journal of Mathematics, 2004

Let X be a Riemann surface of genus g. The surface X is called elliptichyperelliptic if it admits a conformal involution h such that the orbit space X~ (h) has genus one. The involution h is then called an elliptichyperelliptic involution. If g > 5 then the involution h is unique, see [A]. We call symmetry to any anticonformal involution of X. Let Aut + (X) be the group of conformal and anticonformal automorphisms of X and let a, T be two symmetries of X with fixed points and such that {a, ha} and {T, hT} are not conjugate in Aut+(X). We describe all the possible topological conjugacy classes of {a, ah, T, rh}. As consequence of our study we obtain that, in the moduli space of complex algebraic curves of genus g (g even > 5), the subspace whose elements are the elliptichyperelliptic 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.

Symmetries of Riemann surfaces with large automorphism group

Mathematische Annalen, 1974

A Riemann surface is symmetric if it admits an anti-conformal involution. The basic question which we discuss in this paper is whether compact Riemann surfaces of genus g > t which admit large groups of automorphisms are symmetric. As is weU-known, the automorphism group of a compact Riemann surface of genus g > 1 is finite and bounded above by 84(g-1). Macbeath ([t21 13]) has found infinitely many g for which this bound is attained. We show that all the surfaces found by Macbeath's methods are indeed symmetric. However, we do exhibit an example of a non-symmetric Riemann surface of genus g = ! 7 which does admit 84(g-1) automorphisms. We also study Riemann surfaces admitting automorphisms of large order. The order of an automorphism of a Riemann surface of genus g is bounded above by 4g + 2 and this bound is attained for every g [8]. We show that all Riemann surfaces admitting automorphisms of order greater that 2g + 2 are symmetric. There is a close link between our work and the theory of irreflexible regular maps on surfaces. (See § 8 for definitions.) There is a connection between the groups of regular maps and large groups of automorphisms of compact Riemann surfaces. Indeed, every group of automorphisms ofa Riemann surface of genus g of order greater than 24(g-1) is also the group of some regular map and conversely, every group of a regular map can be thought of as the group of automorphisms of a Riemann surface. The irreflexible regular maps turn out to be rather exceptional. (In fact, it was suggested in early editions of [3] that they did not exist for surfaces of genus O > 1). We show in the above correspondence that large groups of automorphisms of non-symmetric surfaces will give rise to irreflexible regular maps, but that the converse of this fact is not always true. Thus, for example, groups of automorphisms of order greater than 24(g-1) of a compact non-symmetric Riemann surface of genus g are more exceptional than irreflexible regular maps. There is another interpretation of symmetric Riemann surfaces which is of interest. Every compact Riemann surface can be obtained as the Riemann surface of an algebraic curve f(z, w) = 0. A Riemann surface

Symmetries of real cyclic p-gonal Riemann surfaces

Pacific Journal of Mathematics, 2004

A closed Riemann surface X which can be realised as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. A p-gonal Riemann surface is called real p-gonal if there is an anticonformal involution (symmetry) σ of X commuting with the p-gonal morphism. If the p-gonal morphism is a cyclic regular covering the Riemann surface is called real cyclic p-gonal, otherwise it is called real generic p-gonal. The species of the symmetry σ is the number of connected components of the fixed point set Fix (σ) and the orientability of the Klein surface X/ σ. In this paper we find the species for the possible symmetries of real cyclic p-gonal Riemann surfaces by means of Fuchsian and NEC groups.

On the fixed-point set of automorphisms of non-orientable surfaces without boundary

The Epstein Birthday Schrift

Macbeath gave a formula for the number of fixed points for each non-identity element of a cyclic group of automorphisms of a compact Riemann surface in terms of the universal covering transformation group of the cyclic group. We observe that this formula generalizes to determine the fixed-point set of each non-identity element of a cyclic group of automorphisms acting on a closed non-orientable surface with one exception; namely, when this element has order 2. In this case the fixed-point set may have simple closed curves (called ovals) as well as fixed points. In this note we extend Macbeath's results to include the number of ovals and also determine whether they are twisted or not.

One-dimensional families of Riemann surfaces of genus g with 4\text {g}+4$$ 4 g + 4 automorphims

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017

We prove that the maximal number ag + b of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is 4g + 4. For each integer g ≥ 2 we find an equisymmetric complex-uniparametric family A g of Riemann surfaces of genus g having automorphism group of order 4g + 4. For g ≡ −1mod4 we present another uniparametric family K g with automorphism group of order 4g + 4. The family A g contains the Accola-Maclachlan surface and the family K g contains the Kulkarni surface.