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.     

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

An explicit upper bound for Weil-Petersson volumes of the moduli spaces of punctured Riemann surfaces

An explicit upper bound for the Weil-Petersson volumes of punctured Riemann surfaces is obtained using Penner's combinatorial integration scheme from [4]. It is shown that for a fixed number of punctures n and for genus g increasing, while this limit is exactly equal to two for n=1. Received: 17 May 2000 / Revised version: 9 August 2000 / Published online: 23 July 2001     

Configuration Models For Moduli Spaces of Riemann surfaces with boundary

In this article we consider Riemann surfacesF of genus g ≥ 0 with n ≥ 1 incoming and m ≥ 1 outgoing boundary circles, where on each incoming circle a point is marked. For the moduli space m_g*(m, n) of all suchF of genusg ≥ 0 a configuration space model Rad_h(m, n) is described: it consists of configurations of h = 2g-2+m+n pairs of radial slits distributed over n annuli; certain combinatorial conditions must be satisfied to guarantee the genusg and exactly m outgoing circles. Our main result is a homeomorphism between Rad_h(m, n) and M_g*(m,n). The space Rad_h(m, n) is a non-compact manifold, and the complement of a subcomplex in a finite cell complex. This can be used for homological calculations. Furthermore, the family of spaces Rad_h(m, n ) form an operad, acting on various spaces connected to conformai field theories.     

An infinitesimal trace formula for the Laplace operator on compact Riemann surfaces

We derive an infinitesimal (or variational) version of the Selberg trace formula for compact Riemann surfaces, which gives information on the behaviour of the eigenvalues of the Laplace-Beltrami operator as the surface varies over the appropriate moduli space.     

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g. Partially supported by DGICYT PB 89-201 and Science Plan 910021 Partially supported by DGICYT PB 89/379/C02/01 and Science Plan 910021 Partially supported by DGICYT After the typing of this paper we have heard about a Ph.D. Thesis by Britta Krapp on questions related to the problem studied here.     

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     

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.     

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.     

Relations on the period mapping giving extensions of mixed hodge structures on compact Riemann surfaces

We consider a period map from Teichmüller space to , which is a real vector bundle over the Siegel upper half space. This map lifts the Torelli map. We study the action of the mapping class group on this period map. We show that the period map from Teichmüller space modulo the Johnson kernel is generically injective. We derive relations that the quadratic periods must satisfy. These identities are generalizations of the symmetry of the Riemann period matrix. Using these higher bilinear relations, we show that the period map factors through a translation of the subbundle and is completely determined by the purely holomorphic quadratic periods. We apply this result to strengthen some theorems in the literature. One application is that the quadratic periods, along with the abelian periods, determine a generic marked compact Riemann surface up to an element of the kernel of Johnson's homomorphism. Another application is that we compute the cocycle that exhibits the mapping class group modulo the Johnson kernel as an extension of the group SP _g () by the group .     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

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.     

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.     

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 the geometry of the space of oriented lines of Euclidean space

We prove that the space of all oriented lines of the n-dimensional Euclidean space admits a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions, exactly when n=3 or n=7 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). Up to equivalence, there are two such metrics for each dimension, and they are of split type and complete. Besides, we prove that the given metrics are Kähler or nearly Kähler if n=3 or n=7, respectively.     

The Thurston boundary of Teichmüller space and complex of curves

Let S be a closed orientable surface with genus g?2. For a sequence σi in the Teichmüller space of S, which converges to a projective measured lamination [λ] in the Thurston boundary, we obtain a relation between λ and the geometric limit of pants decompositions whose lengths are uniformly bounded by a Bers constant L. We also show that this bounded pants decomposition is related to the Gromov boundary of complex of curves.     

Semigroup of matrices acting on the max-plus projective space

We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image.