p-Groups acting on Riemann surfaces

Let p be a prime integer and let $r\ge 3$ be an integer so that $p\ge 5r?7$. We show that a closed Riemann surface S of genus $g\ge 2$ has at most one p-group H of conformal automorphisms so that $S/H$ has genus zero and exactly r cone points. This, in particular, asserts that, for $r=3$ and $p\ge 11$, the minimal field of definition of S coincides with that of $(S,H)$. Another application of this fact, for the case that S is pseudo-real, is that $\mathrm{Aut}(S)/H$ must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.     

The gonality of Riemann surfaces under projections by normal coverings

A compact Riemann surface X of genus g≥2 which can be realized as a q-fold, normal covering of a compact Riemann surface of genus p is said to be (q,p)-gonal. In particular the notion of (2,p)-gonality coincides with p-hyperellipticity and (q,0)-gonality coincides with ordinary q-gonality. Here we completely determine the relationship between the gonalities of X and Y for an N-fold normal covering X→Y between compact Riemann surfaces X and Y. As a consequence we obtain classical results due to Maclachlan (1971) [5] and Martens (1977) [6].     

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.     

The topology of balls and Gromov hyperbolicity of Riemann surfaces

We prove that every ball in any non-exceptional Riemann surface with radius less or equal than is either simply or doubly connected. We use this theorem in order to study the hyperbolicity in the Gromov sense of Riemann surfaces. The results clarify the role of punctures and funnels of a Riemann surface in its hyperbolicity.     

On the weights of the fixed points of an automorphism of a compact Riemann surface

In this work we study automorphisms of compact Riemann surfaces with more than four fixed points. We obtain a lower bound for the weight of each of these fixed points. The discussion depends on the parities of the order of the automorphism and the number of fixed points. Moreover, we discuss the sharpness of our bounds. Received: 15 February 2005     

Equations in characteristic 3 with Mathieu groups as their Galois groups

Galois groups of some explicit equations in characteristic 3 will be computed to be the Mathieu group of degree 12. We use the resolution of singularities of plane curves to get a lower bound for the Galois groups. Also we use factorizations of polynomials over the prime field to sharpen the lower bound. Finally we use the linearization process to get an appropriate upper bound for the Galois groups.Partly supported by PRF grant 690-1395-1920.     

Algebraic convergence of finitely generated Kleinian groups in all dimensions

In this paper we prove that for a sequence {G_i,r} of r-generator Kleinian groups acting on , if {G_i,r} satisfies Condition A, then its algebraic limit is also a Kleinian group. This generalizes the main result in [X. Wang, Algebraic convergence theorems of n-dimensional Kleinian groups, Israel J. Math. 162 (2007) 221-233].     

Sets of periods for automorphisms of compact Riemann surfaces

Let G=〈f〉 be a finite cyclic group of order N that acts by conformal automorphisms on a compact Riemann surface S of genus g≥2. Associated to this is a set A of periods defined to be the subset of proper divisors d of N such that, for some x∈S, x is fixed by f^d but not by any smaller power of f. For an arbitrary subset A of proper divisors of N, there is always an associated action and, if g_A denotes the minimal genus for such an action, an algorithm is obtained here to determine g_A. Furthermore, a set A_max is determined for which g_A is maximal.     

Shared values of meromorphic functions on compact Riemann surfaces

Let S be a compact Riemann surface of genus g and gonality d. We derive upper bounds (in terms of g and/or d) for the number of values that two non-constant meromorphic functions on S can share. The case d = 2 (i.e., the surface is hyperelliptic or elliptic) is studied in more detail.Received: 14 April 2004     

Quasilinear -equation on bordered Riemann surfaces

We consider the existence of solutions of a nonlinear Riemann-Hilbert problem for a quasilinear -equation on a bordered Riemann surface. The first author was supported in part by a grant ``Analiza in geometrija' P1-0291 from the Ministry of Higher Education, Science and Technology of the Republic of Slovenia. The second author was supported in part by grants from FEDER y Ministerio de Ciencia y Tecnología numbers BFM2001-3894 MTM 2004-05878 and Consejería de Educacion Cultura y Deportes del Gobierno de Canarias, PI 2001/091.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

Simultaneous metric uniformization of foliations by Riemann surfaces

We consider a two-dimensional linear foliation on torus of arbitrary dimension. For any smooth family of complex structures on the leaves we prove existence of smooth family of uniformizing (conformal complete flat) metrics on the leaves. We extend this result to linear foliations on and families of complex structures with bounded derivatives C ³-close to the standard complex structure. We prove that the analogous statement for arbitrary C two-dimensional foliation on compact manifold is wrong in general, even for suspensions over in dimension 3 the uniformizing metric can be nondifferentiable at some points; in dimension 4 the uniformizing metric of each noncompact leaf can be unbounded.     

Twisted second homology groups of the automorphism group of a free group

In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for the 2-torsion part, using combinatorial group theory.     

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2≤p≤6. We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p≥2.     

Metacyclic groups as automorphism groups of compact Riemann surfaces

Let X be a compact Riemann surface of genus \(g\ge 2\), and let G be a subgroup of \(\mathrm{Aut}(X)\). We show that if the Sylow 2-subgroups of G are cyclic, then \(|G|\le 30(g-1)\). If all Sylow subgroups of G are cyclic, then, with two exceptions, \(|G|\le 10(g-1)\). More generally, if G is metacyclic, then, with one exception, \(|G|\le 12(g-1)\). Each of these bounds is attained for infinitely many values of g.     

Uniqueness of orientation-preserving free actions of finite abelian groups on surfaces

Let G be a finite abelian group. We list all the cases where the topological equivalence class of orientation-preserving free G-actions on a closed surface is unique. Moreover, we obtain the classification of topological equivalence classes of orientation-preserving free actions of finite abelian groups of rank 2.