On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

Riemann surfaces

Since the classical work of Riemann, Plein, Chobe, and Poincaré, in mathematics the interest in the theory of Riemann surfaces and groups has not abated. The present survey covers papers reviewed in RZhMat during the period 1967–1976 primarily in the sections Algebra. Topology. Geometry. The following topics are considered most completely and thoroughly: the topology of Riemann surfaces and their automorphisms, Fuchsian groups, Teichmüller spaces, and spaces of moduli.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 191–245, 1978,     

On embeddings of modular curves in projective spaces

We use explicit results on modular forms (Mui?, Ramanujan J 27:188–208, 2012) via uniformization theory to obtain embeddings of modular curves and more generally of compact Riemann surfaces attached to Fuchsian groups of the first kind in certain projective spaces. We obtain families of embeddings which vary smoothly with respect to a parameter in the upper-half plane. We study local expression for the divisors attached to the maps in the family.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

On Cyclic Groups of Automorphisms of Riemann Surfaces

The question of extendability of the action of a cyclic groupof automorphisms of a compact Riemann surface is considered.Particular attention is paid to those cases corresponding toSingerman's list of Fuchsian groups which are not finitely-maximal,and more generally to cases involving a Fuchsian triangle group.The results provide partial answers to the question of whichcyclic groups are the full automorphism group of some Riemannsurface of given genus g>1.     

Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group — called its commensurability automorphism group, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in $ T_\infty $), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, $ T_\infty $. Now, each point of $ T_\infty $ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is arithmetic. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $ T_\infty $, are studied by us.     

Some bounds for the number of coincidences of morphisms between closed riemann surfaces

We give some bounds for the number of coincidences of two morphisms between given compact Riemann surfaces (complete complex algebraic curves) which generalize well known facts about the number of fixed points of automorphisms. In the particular case in which both surfaces are hyperelliptic, our results permit us to obtain a bound for the number of morphisms between them. The proof relies on the idea, first used by Schwarz in the case of automorphisms, of representing a morphism by its action on the set of Weierstrass points.     

Riemann surfaces with orbifold points

We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we describe necessary and sufficient conditions for the group generated by the Fuchsian group of the surface with added inversions to be of the almost hyperbolic Fuchsian type. All the techniques elaborated for the bordered surfaces (quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebra of geodesic functions) are equally applicable to the surfaces with orbifold points.     

The Geometry of Two Generator Groups: Hyperelliptic Handlebodies

A Kleinian group naturally stabilizes certain subdomains and closed subsets of the closure of hyperbolic three space and yields a number of different quotient surfaces and manifolds. Some of these quotients have conformal structures and others hyperbolic structures. For two generator free Fuchsian groups, the quotient three manifold is a genus two solid handlebody and its boundary is a hyperelliptic Riemann surface. The convex core is also a hyperelliptic Riemann surface. We find the Weierstrass points of both of these surfaces. We then generalize the notion of a hyperelliptic Riemann surface to a hyperelliptic three manifold. We show that the handlebody has a unique order two isometry fixing six unique geodesic line segments, which we call the Weierstrass lines of the handlebody. The Weierstrass lines are, of course, the analogue of the Weierstrass points on the boundary surface. Further, we show that the manifold is foliated by surfaces equidistant from the convex core, each fixed by the isometry of order two. The restriction of this involution to the equidistant surface fixes six generalized Weierstrass points on the surface. In addition, on each of these equidistant surfaces we find an orientation reversing involution that fixes curves through the generalized Weierstrass points.Mathematics Subject Classifications (2000). primary 30F10, 30F35, 30F40; secondary 14H30, 22E40.     

Dynamique des automorphismes des surfaces projectives complexes

We study the dynamics of automorphisms of complex projective surfaces. Letbe such an automorphism whose topological entropy is not zero. We construct a probability measure associated toand the complex structure. This measure is -invariant, ergodic and has maximal entropy. This is the unique measure satisfying these properties and periodic points are equidistributed with respect to this measure.     

Construction et classification de certaines solutions algébriques des systèmes de Garnier

IsomonodromicdeformationsofFuchsianequationsoforder2onRiemann sphere are parameterized by the solutions of Garnier system. The purpose of this paper is to construct algebraic solutions exotic, i.e. corresponding to deformations of Fuchsian equation with Zariski dense monodromy. Specifically, we classify all the algebraic solutions (complete) exotic constructed by the method of pull-back of Doran-Kitaev: they are deduced from the data isomonodromic deformations pulling back a Fuchsian equation E given by a family of branched coverings ? _t . We first introduce the structures and associated orbifoldes underlying Fuchsian equation. This allows us to have are fined version of the Riemann Hurwitz formula that allows us quickly to show that E must be hypergeometric. Then we come to limit the degree of ? and exponents, and finally to Painlevé VI. We explicitly construct one of these solutions.     

On Riemann surfaces with non-unique cyclic trigonal morphism

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called cyclic trigonal Riemann surface. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus greater or equal to 5. Using the characterization of cyclic trigonality by Fuchsian groups given in [3], we obtain the Riemann surfaces of low genus with non-unique trigonal morphisms. Partially supported by BFM2002-4801. Partially supported by the Swedish Research Council (VR)     

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.     

The topological types of some irregular Kähler surfaces

The topological type of generalized Kummer surfaces is described in terms of sphere bundles over Riemann surfaces and the complex projective plane. Explicit examples of sets of pairwise non-diffeomorphic K?hler surfaces of the same topological type are given. Received: 5 January 2000     

The geometry at infinity of a hyperbolic Riemann surface of infinite type

We study geodesics on planar Riemann surfaces of infinite type having a single infinite end. Of particular interest is the class of geodesics that go out the infinite end in a most efficient manner. We investigate properties of these geodesics and relate them to the structure of the boundary of a Dirichlet polygon for a Fuchsian group representing the surface.      

Symmetry types of cyclic covers of the sphere

We examine all compact Riemann surfaces of genus greater than one which admit a cyclic group of automorphisms that yields a covering of the Riemann sphere with exactly three branch points. We determine the number of non-conjugate symmetries of each of these surfaces. For each symmetry, we determine the number of ovals it fixes and whether the orbit space under the symmetry is orientable or not. This yields the species of each symmetry and the symmetry type of each surface. Explicit defining equations of each surface and symmetry are given.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

On symplectic automorphisms of elliptic surfaces acting on CH0

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Auts(S) of symplectic automorphisms acts trivially on the Albanese kernel CH₀(S)_alb of the 0-th Chow group CH₀(S), unless possibly if the geometric genus and the irregularity satisfy p_g(S) = q(S) ∈ {1, 2}. In the exceptional cases, the image of the homomorphism Auts(S) → Aut(CH₀(S)alb) has the order at most 3. Our arguments actually take care of the...