The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…
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48 results for “Hyperelliptic components”
The study finds a unique systole maximum in non-hyperelliptic surfaces.
problem Understanding systole functions on translation surfaces.
method Analyzing local and global maxima of systole functions.
result Local maxima are not global in non-hyperelliptic components.
Classifies Teichmüller curves in specific hyperelliptic components of meromorphic differentials.
problem Classifying Teichmüller curves in hyperelliptic components of meromorphic strata.
method Non-existence criterion based on intersections with moduli space boundary.
result Contradiction to algebraicity of candidate Teichmüller curves outside Hurwitz covers.
In hyperelliptic components, GL(2,R) orbits are either closed, dense, or cover loci.
problem Understanding GL(2,R) orbits in hyperelliptic components of abelian differentials.
method Analyzing orbits in hyperelliptic components of abelian differentials.
result Orbits are either closed, dense, or cover loci.
Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low genera the strata of meromorphic quadratic …
Study pseudo-Anosov homeomorphisms on translation surfaces to compute geodesic lengths.
problem Compute the geodesic lengths on hyperelliptic connected components of Teichmüller space.
method Developed a new framework to study pseudo-Anosov homeomorphisms on translation surfaces.
result Computed the systole of Teichmüller geodesic flow on hyperelliptic components.
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and …
Finite groups with a hyperelliptic involution have a 2-rank of at most 4.
problem Finite groups acting on hyperelliptic 3-manifolds and their sectional 2-rank.
method Analyzing sectional 2-rank of finite groups containing hyperelliptic involutions.
result The sectional 2-rank of such groups is at most 4, with 4 being the best possible upper bound.
Genus Torelli space is the moduli space of genus curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when these com…
The paper classifies translation surfaces in a specific hyperelliptic component and finds the maximum number of disjoint geodesics.
problem Classifying and understanding translation surfaces in a specific hyperelliptic component.
method Analyzing geodesics and Euclidean structures on hyperelliptic surfaces.
result A classification theorem for translation surfaces in and the maximum number of disjoint geodesics.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy growth rate.
Classifies GL(2,R) invariant markings over Abelian differential components.
problem Classifying GL(2,R) invariant point markings over Abelian differential components.
method Analyzing hyperelliptic components and their invariant markings.
result Invariant markings arise from specific points or involution exchanges, and can determine holomorphic sections.
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1math.GT
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundament…
Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropymath.GT
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surfa…
Extends abelian differentials to log twisted differentials with spin and hyperelliptic structures.
problem Compactify the moduli space of abelian differentials with spin and hyperelliptic structures.
method Introduce log twisted differentials and hyperelliptic differentials using stable log maps and admissible covers.
result Proves the existence of up to three connected components in the open strata of log twisted differentials.
Minimal constructions of meanders and hyperelliptic pillowcase covers help in understanding ratio-optimizing pseudo-Anosovs.
problem Understanding ratio-optimizing pseudo-Anosovs in moduli spaces of quadratic differentials.
method Minimal constructions of meanders and hyperelliptic pillowcase covers.
result Existence of ratio-optimizing pseudo-Anosovs deep in the Johnson filtration.
The paper proves that on translation surfaces, regular points are part of infinitely many geodesics with dense directions.
problem Existence and density of geodesics through regular points on translation surfaces.
method Apisa's classifications of periodic points and orbit closures, recent Eskin-Filip-Wright result.
result Regular points on translation surfaces are part of infinitely many geodesics with dense directions.
Finite groups act on 3-manifolds with specific involution properties.
problem Understanding finite groups acting on hyperelliptic 3-manifolds.
method Analyzing finite groups containing hyperelliptic involutions with specific fixed-point sets.
result Simple groups containing hyperelliptic involutions are isomorphic to or four other small groups.
We construct a relation among right-handed Dehn twists in the mapping class group of a compact oriented surface of genus g with 4g+4 boundary components. This relation gives an explicit topological description of 4g+4 disjoint (-1)-sections of a hyperelliptic Lefschetz fibration of genus g on the manifold {CP}^2#(4g+5)…
The paper studies maps on graphs and their fibers, focusing on hyperelliptic graphs.
problem Analyzing maps on graphs and their fibers.
method Analyzes maps Phi and Phi* on Culler--Vogtmann's outer space CV_n and its quotient T_n, focusing on hyperelliptic graphs.
result Fibers of Phi in T_n are aspherical and pi_1-injective subspaces.
According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant o…
Complex manifold describes solvable Pell-Abel equations with fixed degrees.
problem Understanding the space of solvable Pell-Abel equations with fixed degrees.
method Described the space of Pell-Abel equations as a complex manifold and computed its connected components.
result The space of Pell-Abel equations with fixed degrees forms a complex manifold with connected components described by an invariant.
In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved…
The study constructs surfaces to show ratio-optimizing pseudo-Anosovs are common in Abelian differentials.
problem Understanding ratio-optimizing pseudo-Anosovs in Abelian differentials.
method Constructing square-tiled surfaces to demonstrate the ubiquity of ratio-optimizing pseudo-Anosovs.
result Pseudo-Anosovs optimizing the ratio of Teichmüller to curve graph translation length are common in Abelian differentials.
Study shows monodromy kernels are large, failing to prove commensurability in specific strata.
problem Proving commensurability of mapping class groups through monodromy kernels.
method Analyzing monodromy maps for specific strata in translation surfaces.
result Kernels of monodromy maps contain a non-abelian free group of rank 2.
Unique Teichmüller curve found in complex geometry.
problem Classifying Teichmüller curves in complex geometry.
method Complete classification of algebraically primitive Teichmüller curves.
result Veech 14-gon generates unique algebraically primitive Teichmüller curve.
Study flat metrics from right prisms, finding non-lattice surfaces with translation coverings.
problem Analyzing flat metrics from right regular prisms.
method Viewing prisms as n-differentials and analyzing unfoldings, proving translation coverings to hyperelliptic surfaces.
result Non-lattice surfaces admit translation coverings to hyperelliptic surfaces, allowing explicit computation of orbit closures and counting problems.
A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…
Machine learning accurately distinguishes Sato-Tate groups for hyperelliptic curves.
problem Arithmetic of hyperelliptic curves and Sato-Tate conjecture.
method Bayesian classifier and machine learning techniques applied to L-functions of hyperelliptic curves.
result Machine learning can distinguish Sato-Tate groups with high accuracy and speed.
We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
Classifies periodic diffeomorphisms and hyperelliptic involutions on surfaces.
problem Classifying periodic diffeomorphisms and involutions on surfaces.
method Refinement of Ishizaka's result using Dehn twist presentations.
result Dehn twist presentations of hyperelliptic periodic mapping classes.
Explicit solutions found for mean field equations on hyperelliptic curves of genus two.
problem Finding explicit solutions to mean field equations on complex hyperelliptic curves.
method Analyzing the Gaussian curvature function of the canonical metric on hyperelliptic curves of genus two.
result Explicit solutions to mean field equations are determined by the Gaussian curvature function.
We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute the number of con…
Analytic curves linked to algebraic ones via Schottky groups.
problem Moving between analytic and algebraic representations of Riemann surfaces.
method Identifying Riemann surfaces with Schottky groups and constructing families of non-hyperelliptic surfaces.
result Construction of families of non-hyperelliptic surfaces with specific properties.
The Whitham flow for hyperelliptic curves has singularities that can be smoothly extended.
problem Singularities in the Whitham flow for hyperelliptic spectral curves.
method Analysis of deformations preserving periods of a meromorphic differential.
result Stable and unstable manifolds are non-empty, and the flow can be extended through the singularity.
The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by t…
Tool for contracting subcurves of hyperelliptic curves, proving differential implications.
problem Understanding differentials on hyperelliptic curves and their limits.
method Flexible tool for contracting subcurves, proving Gorenstein contractions and dualising bundles.
result Hyperelliptic multiscale differentials determine Gorenstein contractions of nodal curves.
Study of hyperelliptic mapping class groups with applications and profinite completions.
problem Understanding hyperelliptic mapping class groups and their properties.
method Defined and studied hyperelliptic mapping class groups, applied theory to counterexamples, and examined profinite completions.
result Found a counterexample to a conjecture about mapping class groups and extended congruence subgroup property.
The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As…
New Einstein-Weyl spaces derived from hyperelliptic curves.
problem Constructing new Lorentzian Einstein-Weyl spaces.
method Using hyperelliptic curves and minitwistors.
result Spaces are diffeomorphic to 3D deSitter space with closed geodesics.
Algorithms compute the topology of hyperelliptic curves in 2D and 3D.
problem Computing the topology of hyperelliptic curves in higher dimensions.
method Birational mapping of the plane or space to compute the topology of the curve.
result Algorithms implemented in { t Maple} for computing the topology of hyperelliptic curves.
Discrete PU(1,1) representations of hyperelliptic groups are proven.
problem Characterizing PU(1,1) representations of hyperelliptic groups.
method Proving representations are basic if and only if they are discrete and faithful.
result A conjecture by S. Anan'in and E. Bento Gonçalves is partially proven.
Study on hyperelliptic Lefschetz fibrations, finding singular fiber counts.
problem Determining the minimal number of singular fibers in hyperelliptic Lefschetz fibrations.
method Analyzing complex surfaces and their Lefschetz fibrations over the 2-sphere.
result Minimal number of singular fibers is 2g+4 for even g≥4 and 2g+6 for odd g≥7.
We study relations of the Weierstrass's hyperelliptic al-functions over a non-degenerated hyperelliptic curve of arbitrary genus as solutions of sine-Gordon equation using Weierstrass's local parameters, which are characterized by two ramified points. Though the hyperelliptic solutions of the sine-Gord…
Chart descriptions are a graphic method to describe monodromy representations of various topological objects. Here we introduce a chart description for hyperelliptic Lefschetz fibrations, and show that any hyperelliptic Lefschetz fibration can be stabilized by fiber-sum with certain basic Lefschetz fibrations.
We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus , the number of non-separating vanishing cycles and the number of singular fibers satisfy the inequality $N \…
The study finds bounds on homologically independent loops on hyperelliptic hyperbolic surfaces.
problem Finding bounds on the lengths of homologically independent loops on hyperelliptic hyperbolic surfaces.
method Analyzing the genus and using constant upper bounds on minimal length of non-zero period lattice vectors.
result For any , there exists a constant such that every hyperelliptic hyperbolic surface has at least homologically independent loops of length at most .
The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.
problem Understanding and constructing new symplectic 4-manifolds.
method Explicitly constructing Lefschetz pencils on hyperelliptic Lefschetz fibrations.
result Infinite families of symplectic 4-manifolds are diffeomorphic to fiber sums of standard hyperelliptic Lefschetz fibrations.