arXiv research

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g > 1 the order of this group is naturally bounded in terms of g due to a Riemann-Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maxima…

Invariants defined for braid systems under Hurwitz equivalence.

We give a tropical interpretation of Hurwitz numbers extending the one discovered in \cite{CJM}. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.

Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.

Constructs subvarieties in translation surface strata using combinatorial input.

The first eigenvalue of the Laplacian on a unique Hurwitz surface has a sevenfold multiplicity and specific numerical values.

Analogue of classical Hurwitz numbers is defined in the work for regular coverings of surfaces with marked points by seamed surfaces. Class of surfaces includes surfaces of any genus and orientability, with or without boundaries; coverings may have certain singularities over the boundary and marked points. Seamed surfa…

Paper solves the Hurwitz existence problem using fiber products.

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $Δ_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentatio…

The classical Hurwitz numbers of degree n together with the Hurwitz numbers of the seamed surfaces of degree n give rise to the Klein topological field theory. We extend this construction to the Hurwitz numbers of all degrees at once. The corresponding Cardy-Frobenius algebra is induced by arbitrary Young diagrams and …

We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations. In…

We study the Kähler geometry of the classical Hurwitz space $\mathcal{H}^{n,b}$ of simple branched coverings of the Riemann sphere $\mathbb{P}^1$ by compact hyperbolic Riemann surfaces. A generalized Weil-Petersson metric on the Hurwitz space was recently introduced. Deformations of simple branched coverings fit into t…

We discuss the equivalence between the categories of certain ribbon graphs and subgroups of the modular group $Γ$ and use it to construct exponentially large families of not Hurwitz equivalent simple braid monodromy factorizations of the same element. As an application, we also obtain exponentially large families of {\…

Solves a special case of the Hurwitz problem for Riemann surfaces.

New spin on Hurwitz theory connects to Gromov-Witten theory and topological recursion.

Constructs correspondences on hyperelliptic surfaces combining orbifold groups and Blaschke products.

New partial solution to Hurwitz problem for surface branched covers.

In this paper we prove certain Hurwitz equivalence properties in $B_n$. Our main result is that every two Artin's factorizations of $Δ_n ^2$ of the form $H_{i_1} ... H_{i_{n(n-1)}}, \quad F_{j_1} ... F_{j_{n(n-1)}}$ (with $i_k, j_k \in \{1,...,n-1 \}$), where $\{H_1,...,H_{n-1} \}, \{F_1,...,F_{n-1} \}$ are frames, are…

Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of…

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with re…

We continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surfa…

Given two closed orientable surfaces, the Hurwitz existence problem asks whether there exists a branched cover between them having prescribed global degree and local degrees over the branching points. The Riemann-Hurwitz formula gives a necessary condition, which was shown to be also sufficient when the base surface ha…

We define a new Hurwitz problem which is essentially a small core of the simple Hurwitz problem. The corresponding Hurwitz numbers have simpler formulae, satisfy effective recursion relations and determine the simple Hurwitz numbers. We also apply this idea of finding a smaller simpler enumerative problem to orbifold H…

To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…

In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. …

Computes canonical heights for arithmetic log surfaces using Hurwitz zeta function.

Origamis with specific groups have Veech groups that surject onto SL(2, Z/nZ).

Compactifies the space of branched coverings of the sphere.

Develops tropical geometry for weighted Hurwitz numbers, generalizing previous results.

Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In …

Refines Hurwitz numbers with a two-parameter theory.

Classifies orbits of Hurwitz actions on dihedral quandles.

We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function $F$ defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the…

We provide a formula describing the G-module structure of the Hurwitz-Hodge bundle for admissible G-covers in terms of the Hodge bundle of the base curve, and more generally, for describing the G-module structure of the push-forward to the base of any sheaf on a family of admissible G-covers. This formula can be interp…

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible …

For the existence of a branched covering Sigma~ --> Sigma between closed surfaces there are easy necessary conditions in terms of chi(Sigma~), chi(Sigma), orientability, the total degree, and the local degrees at the branching points. A classical problem dating back to Hurwitz asks whether these conditions are also suf…

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…

Study asymptotics of one part monotone Hurwitz numbers in high genus.

We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…

The paper describes a stratification of a compactified Hurwitz space using combinatorial trees.

Minimal dimensions for Riemann surface embeddings computed for specific groups.

In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the or…

The paper explores rational functions with 3 branching points on the Riemann sphere.

Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological r…

Proves transitivity of a specific class of quadratic polynomials.

In this very short note we slightly generalize some relations for one-part double Hurwitz numbers from math.AG/0209282.

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $Δ^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $Δ^{|df|^2}$ and study it as a functional on t…