arXiv research

Study covers of surfaces, showing types and properties.

A branched covering surface-knot is a surface-knot in the form of a branched covering over a surface-knot. For a branched covering surface-knot, we have a numerical invariant called the simplifying number. We show that branched covering surface-knots with degree three have the simplifying numbers less than three.

A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched cover…

A branched covering surface-knot over an oriented surface-knot $F$ is a surface-knot in the form of a branched covering over $F$. A branched covering surface-knot over $F$ is presented by a graph called a chart on a surface diagram of $F$. For a branched covering surface-knot, an addition of 1-handles equipped with cha…

Study covers of Chamanara surface with large Veech groups.

Simply-connected surfaces of general type for n≥5.

The paper studies cyclic covers of rational surfaces and their Hodge structures.

Conditions for simple closed curves in surface covers.

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…

Partial coverings of hyperbolic surfaces equidistribute with geodesics.

The paper characterizes geometrically finite surfaces via geodesic covers.

Random covers of hyperbolic surfaces have a spectral gap with polynomial rate.

Computes constants for cyclic covers of translation surfaces.

Random covers of hyperbolic surfaces follow a specific probability measure.

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…

Study horocycle orbits in $\mathbb{Z}$-covers of hyperbolic surfaces.

Homology of surface coverings solved for genus 3 and above.

The paper studies orbifold splice quotients and log covers of surface pairs.

Tessellations cover planes without gaps or overlaps.

Study homology groups of mapping and Torelli groups for surfaces with abelian covers.

Solve arc diagrams on surfaces via branched covers.

Random covers of surfaces have tangle-free monodromy and the Putman-Wieland property.

We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some cases, we can determine the unknotting numbers.

We introduce a new construction of a surface link in the 4-space. We construct a surface link as a branched covering over the standard torus, which we call a torus-covering link. We show that a certain torus-covering $T^2$-link is equivalent to the split union of spun $T^2$-links and turned spun $T^2$-links. We show th…

A translation structure equips a Riemann surface with a singular flat metric. Not much is known about the shape of a random translation surface. We compute an upper bound on the expected value of the covering radius of a translation surface in any stratum H_1(kappa). The covering radius of a translation surface is the …

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the o…

We obtain several results about stability of the Bergman kernel on a tower of coverings on complex manifolds. An effective version of Rhodes' result is given for a tower of coverings on a compact Riemann surface of genus greater than or equal to 2. Stability of the Bergman kernel is established for towers of coverings …

For a given branched covering between closed connected surfaces, there are several easy relations one can establish between the Euler characteristics of the surfaces, their orientability, the total degree, and the local degrees at the branching points, including the classical Riemann-Hurwitz formula. These necessary re…

We consider the volume entropy of closed flat surfaces of genus $g\geq 2$ and area 1. We show that a sequence of flat surfaces diverges in the moduli space if and only if the volume entropy converges to infinity. Equivalently the Hausdorff dimension of the Gromov boundary of the isometric universal cover tends to infin…

We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing …

For a branched cover between two closed orientable surfaces, the Riemann-Hurwitz formula relates the Euler characteristics of the surfaces, the total degree of the cover, and the total length of the partitions of the degree given by the local degrees at the preimages of the branching points. A very old problem asks whe…

For large genus, precise monodromy groups are calculated for surface covers.

Researchers prove a spectral gap for Hecke covers of Schottky surfaces.

The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.

The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the minimum possible number of vertices of a simplicial complex homotopy equivalent…

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minim…

We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine …

We study the translation surfaces obtained by considering the unfoldings of the surfaces of Platonic solids. We show that they are all lattice surfaces and we compute the topology of the associated Teichmüller curves. Using an algorithm that can be used generally to compute Teichmüller curves of translation covers of p…

The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.

New proof shows surfaces can have identical length spectra but not simple ones.

New method finds unbranched covers with non-kernel homology.

Characterizes compact complex surfaces with finite homotopy rank-sum.

Harmonic maps from surfaces to CAT(k) spheres are branched coverings.

Nonorientable surface mapping class group embeds quasi-isometrically in its orientable cover.

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

The paper characterizes isomorphic covers of surfaces and applies it to distinguish representations.

We prove a generalized version of Kazhdan's theorem for canonical forms on Riemann surfaces. In the classical version, one starts with an ascending sequence $\{S_n \rightarrow S\}$ of finite Galois covers of a hyperbolic Riemann Surface $S$, converging to the universal cover. The theorem states that the sequence of for…

Study smooth linear statistics on random covers of hyperbolic surfaces, showing central limit and variance results.