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…
On-device research index
arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
169,341 papers · 148 categories
Trend · papers per month
48 results for “hyperelliptic mapping class group”
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.
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 .
Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groupsmath.GT
Study identifies roots of hyperelliptic involutions and braid groups in mapping class groups.
problem Identifying roots of hyperelliptic involutions and braid groups in mapping class groups.
method Analyzes braid groups and mapping class groups on surfaces of genus .
result Hyperelliptic involutions have infinitely many square and cubic roots.
Nonorientable surfaces have a faithful linear representation of a specific dimension.
problem Understanding the structure of hyperelliptic mapping class groups of nonorientable surfaces.
method Proved the existence of a faithful linear representation of dimension .
result The hyperelliptic mapping class group of a nonorientable surface of genus has a faithful representation into .
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.
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.
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…
We obtain a simple presentation of the hyperelliptic mapping class group of a nonorientable surface N. As an application we compute the first homology group of with coefficients in .
We consider the hyperelliptic handlebody group on a closed surface of genus . This is the subgroup of the mapping class group on a closed surface of genus consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the …
Study of liftable mapping class group for superelliptic covers.
problem Understanding mapping class groups of superelliptic covers.
method Computational and algebraic methods to study the liftable mapping class group.
result The liftable mapping class group is independent of the degree of the cover and has finite abelianization.
We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from ω-signatures, and show that they are linearly independent in th…
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.
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…
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…
Constructs correspondences on hyperelliptic surfaces combining orbifold groups and Blaschke products.
problem Combining geometric and dynamical properties on hyperelliptic surfaces.
method Analytic combinations on Riemann sphere, algebraic characterization, and Teichmüller spaces.
result Explicit description of correspondences and injection into Hurwitz spaces.
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
problem Analyzing torsion in Ceresa classes of curves.
method Group-theoretic analogues of Johnson/Morita cocycles applied to pro-l etale fundamental groups of curves.
result Example of a non-hyperelliptic curve with torsion Ceresa class.
S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. I…
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 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.
We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that for a fixed genus g there …
Study automorphisms and Torelli groups of right-angled Artin groups.
problem Characterize automorphisms and Torelli groups of right-angled Artin groups.
method Introduce palindromic automorphism and Torelli groups, relate to GL(n,Z) and mapping class groups, find generating sets.
result Finite generating sets for Pi A_G and centralizer, determine when groups coincide.
In this paper, we discuss relations among several invariants of 3-manifolds including Meyer's function, the eta-invariant, the von Neumann rho-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.
Study of lifting maps in branched covers of 3-manifolds, showing non-injectivity.
problem Injectivity of lifting maps in branched covers of 3-manifolds.
method Analysis of branched covers and mapping class groups.
result Lifting map is generally not injective for most regular branched covers of 3-manifolds.
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.
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating cur…
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)…
In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of…
Finite subgroups of good groups correspond to their profinite completions.
problem Characterizing finite subgroups of profinite completions of good groups.
method Proving bijective correspondence between conjugacy classes of finite p-subgroups in and , and analyzing centralizers and normalizers.
result Bijective correspondence between conjugacy classes of finite p-subgroups in and .
The study examines pseudo-Anosov maps on curve complexes and their asymptotic translation lengths.
problem Analyzing the behavior of pseudo-Anosov maps on curve complexes.
method Using sequences of fibers and monodromies in the fibered cone, the asymptotic translation length of pseudo-Anosov maps is studied.
result The asymptotic translation length of pseudo-Anosov maps on curve complexes behaves asymptotically like .
A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}math.GT
We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is t…
Study on second homology group of genus 3 hyperelliptic Torelli group.
problem Understanding the structure of second homology group of genus 3 hyperelliptic Torelli group.
method Analyzing abelian cycles associated with disjoint separating curves and their algebraic properties.
result Simple abelian cycles are linearly independent in the second homology group.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
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…
The paper classifies homomorphisms between braid groups and mapping class groups.
problem Classifying homomorphisms between braid groups and mapping class groups.
method Analyzing specific cases and using surface bundles over configuration spaces.
result Sharp classification of homomorphisms, showing cyclic or standard representations.
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
Formula for signature of handlebody bundles, interpreting cohomology.
problem Signature of handlebody bundles and their monodromy.
method Explicit formula using homological monodromy.
result Cobounding function of Meyer's signature cocycle on handlebody group.
Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is …
The study finds that many mapping class groups have normal generators.
problem Finding normal generators in mapping class groups.
method Provided a criterion for normal closure and applied it to show normal generators for specific classes of mapping classes.
result Many nontrivial periodic mapping classes and pseudo-Anosov mapping classes are normal generators.
For each d>=2, the mapping class group for plane curves of degree d will be defined and it is proved that there exists uniquely the Meyer function on this group. In the case of d=4, using our Meyer function, we can define the local signature for 4-dimensional fiber spaces whose general fibers are non-hyperelliptic comp…
For any positive integer , we exhibit a cofinite subgroup of the mapping class group of a surface of genus at most two such that admits an epimorphism onto a free group of rank . We conclude that has rank at least and the dimension of the second bounded cohomology of each of these ma…
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…
We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid grou…
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
Study vanishing cycles on curves in toric surfaces, identifying specific types of obstructions.
problem Identify vanishing cycles on curves in toric surfaces with specific types of obstructions.
method Investigate monodromy maps and their target space, the mapping class group, to determine vanishing cycles.
result Explicit generators for the Spin mapping class group are provided.
The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifol…
Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.
problem Explicitly describe the Ceresa class for non-hyperelliptic curves.
method Combining algebraic, tropical, and topological perspectives, defining the Ceresa class for curves and surfaces.
result The Ceresa class is torsion in all settings: tropical curves, topological surfaces, and smooth algebraic curves over .
The paper studies mapping class groups of cyclic covers and their liftable counterparts.
problem Understanding the structure of liftable mapping class groups for cyclic covers.
method Analyzes the liftable mapping class groups of regular cyclic covers and derives explicit generating sets.
result The family of self-normalizing subgroups forms an infinite family in .