Rubén A. Hidalgo

SCHOTTKY UNIFORMIZATIONS OF CLOSED RIEMANN SURFACES WITH ABELIAN GROUPS OF CONFORMAL AUTOMORPHISMS

by RUBEN A. HIDALGOf(Received 8 June, 1992)1. Introduction Let us consider a pair (S,H) consisting of a closed Riemannsurface 5 and an Abelian group H of conformal automorphisms of 5. We are interested infinding uniformizations of 5, via Schottky groups, which reflect the action of the group H.A Schottky uniformization of a closed Riemann surface S is a triple (Q, G, ;r.:Q—»5)where G is a Schottky group with Q as its region

On uniqueness of automorphisms groups of Riemann surfaces

Let γ, r, s, ≥ 1 be non-negative integers. If p is a prime sufficiently large relative to the values γ, r and s, then a group H of conformal automorphisms of a closed Riemann surface S of order ps so that S/H has signature (γ, r) is the unique such subgroup in Aut(S). Explicit sharp lower bounds for p in the case (γ, r, s) ∈ {(1, 2, 1), (0, 4, 1)} are provided. Some consequences are also derived.

FIELDS OF MODULI OF CLASSICAL HUMBERT CURVES

The computation of the field of moduli of a closed Riemann surface seems to be a very difficult problem and even more difficult is to determine if the field of moduli is a field of definition. In this paper we consider the family of closed Riemann surfaces of genus five admitting a group of conformal automorphisms isomorphic to

Kleinian groups with common commutator subgroup

{\mathbb Z}_{2}^{4}

Conformal Versus Topological Conjugacy of Automorphisms on Compact Riemann Surfaces

. These surfaces are non-hyperelliptic ones and turn out to be the highest branched abelian covers of the orbifolds of genus zero and five cone points of order two. We compute the field of moduli of these surfaces and we prove that they are fields of definition. This result is in contrast with the case of the highest branched abelian covers of the orbifolds of genus zero and six cone points of order two as there are cases for which the above property fails.

Noded Teichmüller spaces

In this paper, we obtain a certain rigidity property for Kleinian groups. This asserts that if F and G (both non-elementary torsion-free Fuchsian groups) and [F F] = [G G],then F= G (all equalities are taken in the sense of Mobius transformations). This result is connected to Torellis theorem (for closed Riemann surfaces) and can be a step in understanding an equivalent for more general type of Riemann surfaces.

Noded fuchsian groups i

We produce a family of algebraic curves (closed Riemann surfaces) S λ admitting two cyclic groups H 1 and H 2 of conformal automorphisms, which are topologically (but not conformally) conjugate and such that S / H i is the Riemann sphere [Copf ]ˆ. The relevance of this example is that it shows that the subvarieties of moduli space consisting of points parametrizing curves which occur as cyclic coverings (of a fixed topological type) of [Copf ]ˆ need not be normal.

ON THE CONNECTIVITY OF THE BRANCH LOCUS OF THE SCHOTTKY SPACE

LetG be a finitely generated Kleinian group and let Δ be an invariant collection of components in its region of discontinuity. The Teichmüller spaceT(Δ,G) supported in Δ is the space of equivalence classes of quasiconformal homeomorphisms with complex dilatation invariant underG and supported in Δ. In this paper we propose a partial closure ofT(Δ,G) by considering certain deformations of the above hemeomorphisms. Such a partial closure is denoted byNT(Δ,G) and called thenoded Teichmüller space ofG supported in Δ. Some concrete examples are discussed.

SCHOTTKY UNIFORMIZATIONS OF SYMMETRIES

We consider certain type of Kleinian groups called noded Fuchsian groups. Torsion-free noded Fuchsian groups are divided into two families called noded Schottky groups and noded πg groups, respectively. The extended region of discontinuity is defined together a cuspidal topology. We show that these groups uniformize stable Riemann surfaces and generalize some classical results on torsion-free co-compact Fuchsian groups and Schottky groups to the above ones.

Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs

Let M be a handlebody of genus g ≥ 2. The space T (M), that parametrizes marked Kleinian structures on M up to isomorphisms, can be identified with the space MSg of marked Schottky groups of rank g, so it carries a structure of complex manifold of finite dimension 3(g − 1). The space M(M) parametrizing Kleinian structures on M up to isomorphisms, can be identified with Sg, the Schottky space of rank g, and it carries the structure of a complex orbifold. In these identifications, the projection map �: T (M) → M(M) corresponds to the map from MSg onto Sg that forgets the marking. In this paper we observe that the singular locus B(M) of M(M), that is, the branch locus of �, has (i) exactly two connected components for g = 2, (ii) at most two connected components for g ≥ 4 even, and (iii) M(M) is connected for g ≥ 3 odd.