The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
Study covers of surfaces, showing types and properties.
problem Characterize homeomorphism types of surface covers.
method Analyzing homeomorphism types of covers of surfaces, identifying universal covers and homology covers.
result Identify four possible homeomorphism types for certain covers of surfaces.
We study surface links whose link groups are free abelian, and construct various stimulating and highly non-trivial examples of such surface links.
Study homology groups of mapping and Torelli groups for surfaces with abelian covers.
problem Understanding the homology of mapping and Torelli groups for surfaces with specific topologies.
method Examined the first homology group of mapping and Torelli groups with coefficients in the first rational homology group of the universal abelian cover of the surface.
result For surfaces with one boundary component, the twisted homology groups are finite-dimensional, but for surfaces with one puncture, they are infinite-dimensional.
Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypothes…
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …
Characterizes periods of abelian differentials on surfaces.
problem Characterizing periods of abelian differentials in a given stratum.
method Analyzes elements of H1(S,Z,C) to characterize relative periods. result Generalizes previous work on abelian differentials, answering a question by Filip.
Determines conditions for abelian differentials with specific singularities.
problem Finding conditions for abelian differentials with prescribed singularities.
method Analyzes representations of surface fundamental groups and their holonomy.
result Necessary and sufficient conditions for the existence of translation surfaces with prescribed singularities.
Study the boundary of Riemann surfaces with abelian automorphisms.
problem Characterize the boundary of Riemann surfaces with abelian automorphisms.
method Analyze the moduli space and its Deligne-Mumford compactification, focusing on equisymmetric loci.
result Describe the topological strata at the boundary for hyperelliptic and cyclic p-gonal actions. This paper calculates the derivative of surface holonomy for non-abelian gerbes.
problem Calculating the derivative of surface holonomy for non-abelian gerbes.
method Explicit calculation of the derivative formula for surface holonomy of squares mapped into the base manifold.
result Derivation of a formula for the derivative of surface holonomy of squares mapped into the base manifold.
The paper studies components of abelian differentials on surfaces, finding multiple components not seen in unmarked surfaces.
problem Characterizing connected components of strata of abelian differentials.
method Mod r winding numbers and mapping class group-theoretic Euclidean algorithm.
result Found multiple connected components of abelian differentials, distinguished by roots of cotangent bundle.
Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Boundary Dehn twist on K3 surfaces becomes trivial after abelianization.
problem Understanding the boundary Dehn twist on K3 surfaces. method Obstruction from Baraglia-Konno and global Torelli theorem of K3 surfaces. result Boundary Dehn twist becomes trivial after abelianization.
Study shows compact mapping class groups of infinite type surfaces are never perfect.
problem Characterizing the perfection of mapping class groups of infinite type surfaces.
method Analyzing the closure of compactly supported mapping class groups and Torelli groups, examining their abelianizations.
result The abelianization of the closure of compactly supported mapping class groups contains uncountable direct sums of rationals.
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.
Develops a new non-abelian framework for Riemann surfaces and differential equations.
problem Analyzing second-order differential equations on Riemann surfaces.
method Gauge-theoretic framework and non-abelian approach.
result Extends Dedekind's Schwarzian approach to generic one-parameter families of curves of genus g.
Study covers of Chamanara surface with large Veech groups.
problem Characterize finite abelian covers with large Veech groups.
method Analyze Veech groups of finite abelian covers of the Chamanara surface.
result Every free group arises as the projective Veech group of a finite-area infinite translation surface.
Researchers determine the rational abelianization of a subgroup of mapping class groups.
problem Understanding the structure of the Chillingworth subgroup of mapping class groups.
method Using Johnson homomorphism and Casson-Morita homomorphism, they compute the abelianization and order of related Euler classes.
result They find the rational abelianization of the Chillingworth subgroup as a full mapping class group module.
The paper studies orbifold splice quotients and log covers of surface pairs.
problem Understanding orbifold splice quotients and log covers of surface pairs.
method Analyzes orbifold homology and constructs pairs with universal abelian log covers.
result Computes orbifold homology from resolutions and constructs orbifold splice quotients.
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the genus g mapping class group with finite image gives rise to a Kahler extension. T…
This paper shows how to construct Abelian differentials with any prescribed singularities.
problem Constructing Abelian differentials with specific orders and residues.
method Flat representation of Abelian differentials.
result Every pattern of orders and residues can be realized in Abelian differentials, except for two families in genus zero.
We discuss the possibility of lifting finite subgroups, and in particular finite cyclic subgroups, with respect to the canonical projections between automorphism and outer automorphism groups of free groups, surface groups and their abelianizations.
The paper constructs toric vector bundles using spectral networks and non-abelianization.
problem Understanding how holomorphic vector bundles arise from spectral networks and non-abelianization.
method Constructing toric vector bundles on complete toric surfaces via spectral networks and non-abelianization.
result The moduli space of rank 2 toric vector bundles over toric surfaces admits an A-type X-cluster structure. We study Veech surfaces of genus 2 arising from quadratic differentials that are not squares of abelian differentials. We prove that all such surfaces of type (2,2) and (2,1,1) are arithmetic. In (1,1,1,1) case, we reduce the question to abelian differentials of type (2,2) on hyperelliptic genus 3 surfaces with singula…
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…
In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.
Coarse density of subspaces in moduli space of Riemann surfaces.
problem Characterizing subspaces of moduli space that are coarsely dense.
method Using Teichmüller metric and projections of orbit closures in abelian differentials.
result Projections of certain strata in abelian differentials are coarsely dense in moduli space.
Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.
problem Study of Abelian differentials and their geometric properties.
method Associate flat surfaces to Abelian differentials and analyze their families under GL2+(R) action. result Properties of orbit of Abelian differentials under Teichmüller dynamics.
Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. Formula found for maximal systole of hyperbolic surfaces with largest S3 symmetry.
problem Finding the maximal systole of hyperbolic surfaces with largest S3 extendable abelian symmetry. method Derived a formula involving L and K to calculate the maximal systole. result The maximal systole is given by 2arccoshK. There are few known computable examples of non-abelian surface holonomy. In this paper, we give several examples whose structure 2-groups are covering 2-groups and show that the surface holonomies can be computed via a simple formula in terms of paths of 1-dimensional holonomies inspired by earlier work of Chan Hong-Mo…
A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class…
New method for constructing real algebraic surfaces from complex tropical hypersurfaces.
problem Constructing real algebraic surfaces from complex tropical hypersurfaces.
method Non-Abelian patchworking theory.
result The method reproduces all existing isotopy types of surfaces up to degree three.
We construct some non-arithmetic ball quotients as branched covers of a quotient of an Abelian surface by a finite group, and compare them with lattices that previously appear in the literature. This gives an alternative construction, which is independent of the computer, of some lattices constructed by the author with…
Generalizes abelianization for framed local systems over surfaces.
problem Understanding framed local systems over punctured surfaces for various groups.
method Analysis of spectral networks, triangulations, and matrix reinterpretation of path lifting rules.
result Parametrizations of moduli spaces of decorated and framed local systems.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Deformations of compact Riemann surfaces are considered using a Čech cohomology sliding overlaps approach. Cocycles are calculated for conformal cutting and regluing deformations at zeros of Abelian differentials. A second order deformation expansion is presented for the Riemann period matrix. A complete deformation ex…
The article calculates the minimal model dimensions for classifying spaces of surface braid groups.
problem Classifying spaces for families of virtually abelian subgroups of surface braid groups.
method Analyzes the pure and full braid groups of surfaces with at least one boundary component or one puncture, proving analogous results for amenable subgroups.
result Minimal dimensions of models for classifying spaces are equal to the virtual cohomological dimension plus the rank of virtually abelian subgroups.
New representation theory for surface groups to SO0(2,3).
problem Understanding representations of surface groups into special orthogonal groups.
method Non-maximal Anosov representations via Higgs bundles and non-Abelian Hodge correspondence.
result Generalization of Filip's result on weight 3 variation of Hodge structures.
Study shows complex K3 surfaces have infinite free abelian subgroup in their diffeomorphism group.
problem Understanding the structure of diffeomorphism groups of complex K3 surfaces.
method Used families of Seiberg-Witten invariants and moduli spaces of Einstein metrics.
result Proved the existence of a free abelian subgroup of countably infinite rank in the identity component of the diffeomorphism group.
The paper describes geometrically how certain groups act on surfaces.
problem Understanding the geometric structure of virtual Schottky groups.
method Geometric structural decomposition of virtual Schottky groups.
result Provides a geometrical structural decomposition for specific virtual Schottky groups.
It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a k-component 2-link group (k>1) is not free abelian. In this paper, we give examples of T2-links each of whose link groups is a free abelian group of rank three or four. Concerning the T2-l…
The Seiberg-Witten equations are defined on certain complex line bundles over smooth oriented four manifolds. When the base manifold is a complex Kahler surface, the Seiberg-Witten equations are essentially the Abelian vortex equations. Using known non-abelian generalizations of the vortex equations as a guide, we expl…
Refines Haupt's theorem for surface characters.
problem Characterization of holomorphic 1-forms on surfaces.
method Refines Haupt's theorem with divisor data.
result Necessary and sufficient conditions for surface characters.
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
New abelian cohomology theory applied to rigidity of flows.
problem Rigidity of Anosov flows and negatively curved surfaces.
method Development of abelian Livshits theory for transitive Anosov flows.
result Abelian Livshits theorem for homologically full Anosov flows.