Compact Riemann Surface Of Genus Research Articles

Riemann surfaces and restrictively-marked hypermaps

If S is a compact Riemann surface of genus g > 1 then S has at most 84( g − 1) (orientation preserving) automorphisms (Hurwitz). On the other hand, if G is a group of automorphisms of S and | G | > 24( g − 1) then G is the automorphism group of a regular oriented map (of genus g ) and if | G | > 12(g − 1) then G is the automorphism group of a regular oriented hypermap of genus g (Singerman). We generalise these results and prove that if | G | > g − 1 then G is the automorphism group of a regular restrictedly-marked hypermap of genus g . As a special case we also show that a marked finite transitive permutation group (Singerman) is a restrictedly-marked hypermap with the same genus.

On the connectedness of the branch locus of the moduli space of Riemann surfaces

The moduli space $$ \mathcal{M}_g $$ of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus $$ \mathcal{B}_g $$ . In this article we present some results related with the topology of $$ \mathcal{B}_g $$ . We study the connectedness of $$ \mathcal{B}_g $$ for g ≤ 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces consIDered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

The Asymptotic Schottky Problem

Let $${\mathcal{M}_g}$$ denote the moduli space of compact Riemann surfaces of genus g and let $${\mathcal{A}_g}$$ be the moduli space of principally polarized abelian varieties of dimension g. Let $${J : \mathcal{M}_g \rightarrow \mathcal{A}_g}$$ be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image $${\mathcal{J}_g := J(\mathcal{M}_g)}$$ is contained in a proper subvariety of $${\mathcal{A}_g}$$ when g ≥ 4. The classical and long-studied Schottky problem is to characterize the Jacobian locus $${\mathcal{J}_g}$$ in $${\mathcal{A}_g}$$ . In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does $${\mathcal{J}_g}$$ look like “from far away”, or how “dense” is $${\mathcal{J}_g}$$ in the sense of coarse geometry? The large scale geometry of $${\mathcal{A}_g}$$ is encoded in its asymptotic cone, $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ , which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus $${\mathcal{J}_g}$$ is “coarsely dense” in $${\mathcal{A}_g}$$ , which implies that the subset of $${{\rm Cone}_\infty(\mathcal{A}_g)}$$ determined by $${\mathcal{J}_g}$$ actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in $${\mathcal{A}_g}$$ as well. We also study the boundary points of the Jacobian locus $${\mathcal{J}_g}$$ in $${\mathcal{A}_g}$$ and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of $${\mathcal{J}_g}$$ in these compactifications is “small”.

On the connectedness of the branch locus of the moduli space of Riemann surfaces

The moduli space $$ \mathcal{M}_g $$ of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus $$ \mathcal{B}_g $$ . In this article we present some results related with the topology of $$ \mathcal{B}_g $$ . We study the connectedness of $$ \mathcal{B}_g $$ for g ≤ 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces consIDered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

An update on Hurwitz groups

A Hurwitz group is any non-trivial finite quotient of the (2, 3, 7) triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x2 = y3 = (xy)7 = 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that |G| = 84(g – 1), which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc.23 (1990).

The congruence subgroup property for the hyperelliptic modular group: the open surface case

Let $\cM_{g,n}$ and $\cH_{g,n}$, for $2g-2+n>0$, be, respectively, the moduli stack of $n$-pointed, genus $g$ smooth curves and its closed substack consisting of hyperelliptic curves. Their topological fundamental groups can be identified, respectively, with $\GG_{g,n}$ and $H_{g,n}$, the so called Teichmüller modular group and hyperelliptic modular group. A choice of base point on $\cH_{g,n}$ defines a monomorphism $H_{g,n}\hookra\GG_{g,n}$. Let $S_{g,n}$ be a compact Riemann surface of genus $g$ with $n$ points removed. The Teichmüller group $\GG_{g,n}$ is the group of isotopy classes of diffeomorphisms of the surface $S_{g,n}$ which preserve the orientation and a given order of the punctures. As a subgroup of $\GG_{g,n}$, the hyperelliptic modular group then admits a natural faithful representation $H_{g,n}\hookra\out(\pi_1(S_{g,n}))$. The congruence subgroup problem for $H_{g,n}$ asks whether, for any given finite index subgroup $H^\ld$ of $H_{g,n}$, there exists a finite index characteristic subgroup $K$ of $\pi_1(S_{g,n})$ such that the kernel of the induced representation $H_{g,n}\ra\out(\pi_1(S_{g,n})/K)$ is contained in $H^\ld$. The main result of the paper is an affirmative answer to this question for $n\geq 1$.

Chen–Ruan cohomology of some moduli spaces of parabolic vector bundles

Let ( X , D ) be an ℓ-pointed compact Riemann surface of genus at least two. For each point x ∈ D , fix parabolic weights ( α 1 ( x ) , α 2 ( x ) ) such that ∑ x ∈ D ( α 1 ( x ) − α 2 ( x ) ) < 1 / 2 . Fix a holomorphic line bundle ξ over X of degree one. Let P M ξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights { ( α 1 ( x ) , α 2 ( x ) ) } x ∈ D . The group of order two line bundles over X acts on P M ξ by the rule E ∗ ⊗ L ↦ E ∗ ⊗ L . We compute the Chen–Ruan cohomology ring of the corresponding orbifold.

Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula

Let $w$ be an Abelian differential on compact Riemann surface of genus $g\geq 1$. We obtain an explicit holomorphic factorization formula for $\zeta$-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy $|w|^2$, generalizing the classical Ray-Singer result in $g=1$.

Fundamental groups of moduli stacks of stable curves of compact type

Let $\widetilde{\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\widetilde{\cal M}_{g,n}$. Let $\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $K_{g,n}$ be the normal subgroup of $\Gamma_{g,n}$ generated by Dehn twists along separating circles on $S_{g,n}$. As a first application of the above theory, a characterization of $K_{g,n}$ is given for all $n\geq 0$ (for $n=0,1$, this was done by Johnson). Let then ${\cal T}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation $\Gamma_{g,n}\ra Sp_{2g}(Z)$. The abelianization of ${\cal T}_{g,n}$ is determined for all $g\geq 1$ and $n\geq 1$, thus completing classical results by Johnson and Mess.

Reflections of regular maps and Riemann surfaces

A compact Riemann surface of genus g is called an M-surface if it admits an anti-conformal involution that fixes g+1 simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus g ϯ 1 there is a unique M-surface of genus g that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix g curves.

On the rotation angles of a finite subgroup of a mapping class group

Let $G$ be a finite subgroup of the mapping class group of genus $\sigma$, which acts on a compact Riemann surface of genus $\sigma$. In this paper, we introduce a new method to determine the rotation angle of an element $g\in G$ around the fixed points of $g$. Our main result is Theorem 3.2.

The Weil–Petersson geometry of the moduli space of Riemann surfaces

In 2007, Z. Huang showed that in the thick part of the moduli space M g of compact Riemann surfaces of genus g, the sectional curvature of the Weil-Petersson metric is bounded below by a constant depending on the injectivity radius, but independent of the genus g. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichmuller space equipped with a Hilbert structure induced by the Weil-Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.

The mapping class group and the Meyer function for plane curves

For each d ≥ 2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d = 4, using our Meyer function, we can define the local signature for four-dimensional fiber spaces whose general fibers are non-hyperelliptic compact Riemann surfaces of genus 3. Some computations of our local signature will be given.

Non-Abelian Vortices on Riemann Surfaces: an Integrable Case

We consider U(n + 1) Yang–Mills instantons on the space Σ × S 2, where Σ is a compact Riemann surface of genus g. Using an SU(2)-equivariant dimensional reduction, we show that the U(n + 1) instanton equations on Σ × S 2 are equivalent to non-Abelian vortex equations on Σ. Solutions to these equations are given by pairs (A,ϕ), where A is a gauge potential of the group U(n) and ϕ is a Higgs field in the fundamental representation of the group U(n). We briefly compare this model with other non-Abelian Higgs models considered recently. Afterwards we show that for g > 1, when Σ × S 2 becomes a gravitational instanton, the non-Abelian vortex equations are the compatibility conditions of two linear equations (Lax pair) and therefore the standard methods of integrable systems can be applied for constructing their solutions.

Finite Degree Holomorphic Covers of Compact Riemann Surfaces

A conjecture of Ehrenpreis states that any two compact Riemann surfaces of genus at least two have finite degree unbranched holomorphic covers that are arbitrarily close to each other in moduli space. Here we prove a weaker result where certain branched covers associated with arithmetic Riemann surfaces are allowed, and investigate the connection of our result with the original conjecture.

Riemann surfaces of genus g with an automorphism of order p prime and p &gt; g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameterization space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations in $$\mathbb{C}^{2}$$ for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.

Profinite Teichmüller theory

AbstractFor 2g – 2 + n &gt; 0, let Γg, n be the Teichmüller group of a compact Riemann surface of genus g with n points removed Sg, n , i.e., the group of homotopy classes of diffeomorphisms of Sg, n which preserve the orientation of Sg, n and a given order of its punctures. There is a natural faithful representation Γg, n → Out(π 1(Sg, n )). For any given finite index subgroup Γλ of Γg, n , the congruence subgroup problem asks whether there exists a finite index characteristic subgroup K of π 1(Sg, n ) such that the kernel of the induced representation Γg, n → Out(π 1(Sg, n )/K ) is contained in Γλ . The main result of the paper is an affirmative answer to this question. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Geometric quantization of the moduli space of the self-duality equations on a Riemann surface

The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin showed that the moduli space M of solutions of the self-duality equations on a compact Riemann surface of genus g > 1 has a hyper-Kahler structure. In particular M is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which are missing in Hitchin's paper. Next we apply Quillen's determinant line bundle construction to show that M admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above.

Non-completeness of the Arakelov-induced metric on moduli space of curves

Let X be a compact Riemann surface of genus g>1. We study two different, naturally defined metric forms on X: The hyperbolic metric form μhyp,X, arising from hyperbolic geometry, and the Arakelov metric form μAr,X, arising from arithmetic algebraic geometry. Now consider a sequence Xt of Riemann surfaces approaching the Deligne-Mumford boundary of the moduli space Open image in new window of compact Riemann surfaces of genus g. We prove here that Open image in new window

AUTOMORPHISM GROUPS OF RIEMANN SURFACES OF GENUS p+1, WHERE p IS PRIME

We show that if $\cal S$ is a compact Riemann surface of genus $g = p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|\geq\lambda(g-1)$ for some real number $\lambda>6$, then for all sufficiently large $p$ (depending on $\lambda$), $\cal S$ and $G$ lie in one of six infinite sequences of examples. In particular, if $\lambda=8$ then this holds for all $p\geq 17$ and we obtain the largest groups of automorphisms of Riemann surfaces of genenera $g=p+1$.