We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.
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Totally Symmetric Setsmath.GR
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.
problem Conditions for torsion elements to generate symmetric or alternating subgroups.
method Analyzes mapping class groups of surfaces, derives necessary and sufficient conditions for conjugates of torsion elements to generate symmetric or alternating subgroups.
result Symmetric or alternating subgroups cannot contain irreducible mapping classes and hyperelliptic involutions.
Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.
New proofs show smallest non-cyclic quotients for braid and mapping class groups.
problem Classifying smallest non-cyclic quotients of braid and mapping class groups.
method Elementary proofs without Bertrand-Chebyshev theorem.
result Smallest non-cyclic quotients for braid and mapping class groups identified.
We study Question 7.9 in the paper "Monoids in the mapping class group" by Etnyre and Van Horn-Morris; whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid. We answer affirmatively for mapping classes satisfying certain cyclic conditions.
Classifies covering spaces and braid group embeddings in mapping class groups.
problem Classifying covering spaces and their associated mapping class groups.
method Using Birman-Hilden theorem and fundamental groupoid action.
result Constructs infinite families of non-geometric braid group embeddings.
For any smooth compact manifold of dimension at least two we prove that the classifying spaces of its group of diffeomorphisms which fix a set of points or embedded disks (up to permutation) satisfy homology stability. The same is true for so-called symmetric diffeomorphisms of connected sum with co…
Paper computes rational cohomology of spin hyperelliptic mapping class groups.
problem Computing rational cohomology of spin hyperelliptic mapping class groups.
method Computes the -invariant part of the rational cohomology of the pure braid group.
result Includes rational cohomology of spin hyperelliptic mapping class groups of genus .
The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular…
Study quotients of curve complex actions by mapping class group.
problem Understanding actions of mapping class group on curve complex quotients.
method Cone off uniformly quasi-convex subspaces to form symmetric curve sets, non-maximal train track sets, and compression body disc sets. Analyze actions of mapping class group on these quotients.
result Actions of mapping class group on quotients are strongly WPD, non-elementary, and have infinite diameter.
Given a surface of higher genus, we will look at the Weil-Petersson completion of the Teichmuller space of the surface, and will study the isometric action of the mapping class group on it. The main observation is that the geometric characteristics of the setting bear strong similarities to the ones in semi-simple Lie …
New mapping class group actions on Hochschild complexes for modular categories.
problem Understanding actions of mapping class groups on Hochschild complexes of modular categories.
method Construction of a symmetric monoidal functor with excision property.
result Homotopy coherent projective action of mapping class groups on Hochschild complexes.
The paper studies power subgroups of Dehn twists in hyperelliptic mapping class groups.
problem Investigating the index of power subgroups in mapping class groups and hyperelliptic mapping class groups.
method Using a projective representation of mapping class groups through the Kauffman bracket skein module.
result The normal closure of the fifth power of a half-twist has infinite index in the mapping class group of a 2n-punctured sphere.
Einstein solvmanifolds embedded in symmetric spaces.
problem Finding faithful representations of Lie groups as isometric embeddings.
method Investigating special maps for solvable Lie groups, including Einstein and Ricci soliton solvmanifolds.
result All Einstein solvmanifolds can be realized as submanifolds of a symmetric space.
We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms …
Given a family of groups admitting a braided monoidal structure (satisfying mild assumptions) we construct a family of spaces on which the groups act and whose connectivity yields, via a classical argument of Quillen, homological stability for the family of groups. We show that stability also holds with both polynomial…
Study Picard groups of curves with symmetry, focusing on abelian groups and hyperelliptic curves.
problem Understanding the Picard groups of moduli spaces of curves with symmetry.
method Theory of symmetric mapping class groups, finitely generated Picard groups computation.
result Finitely generated Picard groups for moduli spaces of curves with abelian automorphisms.
Study on cohomology of spin hyperelliptic mapping class groups.
problem Cohomology of spin hyperelliptic mapping class groups.
method Study of -invariant part of rational cohomology of pure braid groups.
result Independence of cohomology dimensions in low degrees and formulas for dimensions.
Study block mapping class groups and their finiteness properties.
problem Investigate the finiteness properties of block mapping class groups.
method Consider Cantor surfaces and block mapping class groups with local action prescribed by subgroups.
result Prove finiteness properties of block mapping class groups for spheres and tori.
In 1992, Hitchin used his theory of Higgs bundles to construct an important family of representations of the fundamental group of a closed, oriented surface of genus at least two into the split real form of a complex adjoint simple Lie group. These Hitchin representations comprise a component of the space of conjugacy …
Involutive Hopf monoids yield surface invariants.
problem Involutive Hopf monoids in symmetric monoidal categories.
method Construction of invariants via (co)equalizers and images.
result Categorical generalization of quantum double models.
Study automorphisms of profinite mapping class groups for surfaces with negative Euler characteristic.
problem Determine automorphism groups of profinite completions of mapping class groups.
method Analyzes the structure of automorphism groups and embeddings of profinite Grothendieck-Teichmüller groups.
result Provides natural isomorphisms and embeddings for specific cases of surfaces.
Twisted local wild mapping class groups: configuration spaces, fission trees and complex braidsmath.AG
Study of wild mapping class groups on complex reflection groups.
problem Understanding deformations of wild Riemann surfaces.
method Construction of configuration spaces and combinatorial fission forests.
result Sharp parameterisation of admissible deformation classes of wild Riemann surfaces.
Study on fixed points of random permutations with surface group constraints.
problem Understanding fixed points of random permutations with surface group constraints.
method Computing expected number of fixed points using word maps and surface group constraints.
result The expected number of fixed points is bounded by for shortest representatives.
We use filtrations of the Grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a Riemann surface, and so all harmonic maps from the 2-sphere, to the unitary group for a general class of factorizations by unitons. We show how these specialize to give explicit formu…
The article explores symmetric maps on surfaces, focusing on semi-equivelar maps.
problem Identifying and classifying semi-equivelar maps on surfaces with specific Euler characteristics.
method Analyzing automorphisms and symmetry groups of maps on higher genus surfaces.
result There are at least 39 types of semi-equivelar maps on surfaces with Euler characteristic -2m, m ≥ 2, with symmetry groups isomorphic to dihedral or cyclic groups.
Researchers compute abelianizations of mapping class groups for odd primes.
problem Computing abelianizations of mapping class groups for odd primes.
method Analyzing coverings and associated groups, computing abelianizations.
result Abelianizations exhibit a splitting phenomenon different from .
Study the geometric properties of skew symmetric matrices and orthogonal groups.
problem Understanding the geometric properties of skew symmetric matrices and orthogonal groups.
method Investigate the differential-geometric properties of the exponential map and Riemannian structure.
result Connections between skew symmetric matrices and orthogonal groups are revealed.
Harmonic maps between symmetric spaces have duals.
problem Understanding harmonic maps between symmetric spaces.
method Constructing dual harmonic maps using potentials.
result Duality theorem for harmonic maps into inner symmetric spaces.
Loop group method varies with base point choice.
problem Dependence of loop group method on base point choice.
method Analyzes how loop group method for harmonic maps varies with base point.
result Loop group method results depend on base point selection.
Enumerating Isotopy Classes of Tilings guided by the symmetry of Triply-Periodic Minimal Surfacesmath.GT
New method for counting distinct tilings with symmetrical surfaces.
problem Counting distinct tilings with symmetrical surfaces.
method Deriving representations of mapping class groups and describing tilings as decorations on orbifolds.
result Explicit enumeration of isotopically distinct tilings.
Let be a complex Lie group and denote the group of maps from the unit circle into , of a suitable class. A differentiable map from a manifold into , is said to be of \emph{connection order } if the Fourier expansion in the loop parameter of the -family …
New forms of symmetric shift-invariant subspaces found for harmonic maps.
problem Understanding harmonic maps into symmetric and -symmetric spaces.
method Imposing a symmetry condition on shift-invariant subspaces of a Hilbert space.
result Obtained new general forms for symmetric shift-invariant subspaces and extended solutions.
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of .
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
Unified mathematical framework for non-compact symmetric spaces with negative curvature.
problem Understanding and organizing non-compact symmetric spaces with negative curvature.
method Unified mathematical framework, including explicit distance functions and universality classes.
result Unified classification of non-compact symmetric spaces with non-compact rank r<5.
Unimodular classification of symmetric matrix map-germs.
problem Classifying symmetric matrix map-germs under volume-preserving equivalence.
method Introducing symmetrical quasi-homogeneity and volume-preserving equivalence.
result All simple -equivalence classes coincide with volume-preserving equivalence classes.
The automorphisms of free groups with boundaries form a family of groups A_{n,k} closely related to mapping class groups, with the standard automorphisms of free groups as A_{n,0} and (essentially) the symmetric automorphisms of free groups as A_{0,k}. We construct a contractible space L_{n,k} on which A_{n,k} acts wit…
Survey on real forms of a complex equation and their connection to surface theory.
problem Describing real forms of the complex -Toda equation and their geometric implications.
method Analyzing the integrability of Maurer-Cartan forms for different real forms of loop groups.
result Each real form of corresponds to a specific surface class with integrable frames.
We prove that rationally essential manifolds with suitably large fundamental groups do not admit any maps of non-zero degree from products of closed manifolds of positive dimension. Particular examples include all manifolds of non-positive sectional curvature of rank one and all irreducible locally symmetric spaces of …
Let M be a manifold endowed with a symmetric affine connection The aim of this paper is to describe a quantization map between the space of second-order polynomials on the cotangent bundle T^{*} M and the space of second-order linear differential operators, both viewed as modules over the group of diffeomorphisms …
We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…
Geodesic completeness proven for certain symmetric spaces.
problem Geodesic completeness of compact locally symmetric pseudo-Riemannian manifolds of signature (2,2).
method Analysis of unique parallel lightlike vector fields and group actions.
result Geodesic completeness established for compact locally symmetric spaces modeled on specific spaces.
A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with h…
Study quasi-isometric embeddings in symmetric spaces and Lie groups.
problem Understanding embeddings between symmetric spaces and Lie groups.
method Decompose embeddings into irreducible components and analyze examples.
result Rigidity results extended to semisimple Lie groups, including counterexamples.
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We a…