The study examines modular fusion categories with trivial Torelli group actions.
problem Characterizing modular fusion categories with trivial Torelli group actions.
method Analyzing the mapping class group representations and their kernels.
result For modular fusion categories, the Torelli group is contained in the kernel of the genus-g representation if and only if the category is pointed. We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even component of the space of flat PSL(2,C) connections. The non-trivial part consists …
The paper calculates the top homology group of a specific Torelli group.
problem Computing the top homology group of the genus 3 Torelli group.
method Using group cohomology and representation theory, the authors prove an isomorphism and construct generators.
result An explicit set of generators and relations for the group H_4(I_3, Z) is constructed.
Counterexamples found for Torelli groups of Kähler and hyper-Kähler manifolds.
problem Torelli groups of Kähler and hyper-Kähler manifolds are not always finite.
method Construction of homomorphisms and analysis of diffeomorphisms' actions.
result Found counterexamples showing Torelli groups are not always finite.
Proves Torelli group action is ergodic on Lie group character varieties.
problem Ergodicity of Torelli group action on compact character varieties.
method Proof based on properties of connected, semi-simple Lie groups.
result Torelli group action on compact character varieties is ergodic.
Study of Torelli group action on complex of cycles yields infinite generation result.
problem Determine if genus 3 Torelli group is finitely presented.
method Use spectral sequence for action on complex of cycles to study second homology group.
result Prove that term E0,23 of spectral sequence is infinitely generated. Study Torelli subgroups of handlebody groups and their cohomology.
problem Understanding the cohomology of handlebody Torelli groups.
method Introduce Torelli subgroups, use Johnson homomorphisms, and symplectic representations.
result Describe cup products in the first rational cohomology groups of handlebody Torelli groups.
Proves properties of Torelli Lie algebra for surfaces.
problem Properties of Torelli Lie algebra for surfaces.
method Proves two theorems about Malcev Lie algebra associated to Torelli group.
result Stable Koszul property and trivial Johnson homomorphism kernel.
For a oriented genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F" where F=pi_1(S) and F" is the second term of the derived series. …
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebrai…
Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
problem Understanding symplectic Torelli groups of rational surfaces.
method Using positivity condition, type of cohomology class, and Lagrangian spherical classes.
result Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.
Paper proves Rohlin invariant's uniqueness and extends homology sphere invariants.
problem Proving the uniqueness of Rohlin invariant and extending homology sphere invariants.
method Using the Rohlin invariant's uniqueness, the paper extends invariants from trivial 2-cocycles to those with 2-torsion.
result Generalized invariants of homology spheres with 2-torsion values.
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…
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…
Calculates Dehn twist actions on conformal blocks for modular categories.
problem Order of Dehn twists on spaces of conformal blocks.
method Quantum representations of mapping class groups, ribbon twists, monoidal powers.
result Order of Dehn twists generalizes previous results for small quantum group.
Study of Torelli groups on infinite-type surfaces, focusing on generation and commensuration.
problem Understanding Torelli groups on surfaces of infinite topological type.
method Topological generation and abstract commensuration analysis.
result Abstract commensurator group of Torelli group coincides with mapping class group for infinite-type surfaces.
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.
We introduce machinery to allow ``cut-and-paste''-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a categor…
Complexity of homomorphisms from 3-manifolds to simple groups is #P-complete.
problem Computing homomorphisms from 3-manifolds to simple groups.
method Reduction to circuit evaluation and topological quantum computation.
result Homomorphism problems are #P-complete and NP-complete.
Study on torsion in homology of Torelli group for surfaces.
problem Torsion in homology of Torelli group for surfaces.
method Analysis of abelian cycles and Dehn twists.
result Subgroup of homology generated by abelian cycles is a finite-dimensional Z/2Z-vector space for k=2 and g≥4. Paper tackles extension problem for surface diffeomorphisms.
problem Determining when a subsurface diffeomorphism can be extended to a whole surface.
method Analyzes two homology groups and a difference map for extensions.
result Extension problem depends on subsurface position but in a controlled way.
Two curves' bounding pair maps generate a free group if invariant under an involution.
problem Understanding the Torelli group and its subgroups.
method Analyzing bounding pair maps and their invariance under involution.
result Bounding pair maps generate a free group if invariant under an involution.
For a closed surface S, its Torelli group I(S) is the subgroup of the mapping class group of S consisting of elements acting trivially on H1(S;Z). When S is orientable, a generating set for I(S) is known. In this paper, we give a normal generating set of I(Ng) for …
Study subgroup actions on mapping class groups using Heisenberg representations.
problem Untwisting representations of mapping class groups on Heisenberg subgroups.
method Restrict and analyze twisted representations of mapping class groups to Heisenberg subgroups.
result Untwisting representations on Torelli group for any Heisenberg representation.
Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for …
For large genus, precise monodromy groups are calculated for surface covers.
problem Calculating precise monodromy groups for large genus surface covers.
method Hodge-theoretic methods, including a generic Torelli theorem with coefficients.
result Precise connected monodromy groups are calculated for large genus surface covers.
Finite groups can't smoothly act on compact circles, except trivially.
problem Can finite groups act faithfully on compact one-manifolds by smooth diffeomorphisms?
method Analyzing right-angled Artin groups and their defining graphs, proving no such action exists for C1+bv diffeomorphisms. result No finite index subgroup of certain mapping class groups can act faithfully by C1+bv diffeomorphisms on a compact one-manifold. 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.
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…
Finite index subgroups of certain groups cannot act faithfully on the circle.
problem Finite index subgroups of specific groups cannot act faithfully on the circle.
method Analyzing C1 actions on the circle for finite index subgroups of mapping class groups, automorphism groups, and outer automorphism groups. result No orientation preserving C1 action of finite index subgroups of these groups on the circle can be faithful. Torelli groups' homology is finitely generated in stable range.
problem Whether the homology groups of Torelli subgroups are finitely generated in stable range.
method Using unipotency condition and Tavgen's theorem.
result Homology groups are finitely generated in stable range.
We examine the Johnson filtration of the (outer) automorphism group of a finitely generated group. In the case of a free group, we find a surprising result: the first Betti number of the second subgroup in the Johnson filtration is finite. Moreover, the corresponding Alexander invariant is a non-trivial module over the…
Boundary Dehn twists become trivial after abelianization.
problem Understanding the behavior of boundary Dehn twists in smooth mapping classes.
method Constructing diffeomorphisms and showing commutators represent boundary Dehn twists.
result Boundary Dehn twists become trivial after abelianization.
Study of Torelli group homology reveals infinite rank subgroups.
problem Prove the existence of infinite rank subgroups in Torelli group homology.
method Spectral sequence for the action of Torelli group on cycles.
result Proves existence of infinite rank subgroups in Torelli group homology for k=2g−3 and k≥3g−6. Researchers found the second homology of Torelli groups for large g.
problem Calculating the second rational homology of Torelli groups.
method Explicit calculations for g≥6. result The second rational homology group is determined for Torelli groups with g >= 6.
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 …
Paper defines and analyzes Chillingworth classes for subsurface Torelli groups.
problem Understanding Chillingworth classes for subsurface Torelli groups.
method Combinatorial description and proof of properties of the Chillingworth class.
result Relates Chillingworth class to partitioned Johnson homomorphism.
Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the c…
Torelli group's second homology is finite for large genus surfaces.
problem Determining the finiteness of Torelli group's homology.
method Analyzing rational homology groups of Torelli groups for surfaces of genus g ≥ 51.
result Second rational homology of Torelli group is finite for large genus surfaces.
Study shows Torelli groups are exponentially distorted in mapping class groups with boundary.
problem Distortion of Torelli groups in mapping class groups with boundary.
method Used Broaddus-Farb-Putman's techniques to prove exponential distortion.
result Torelli groups are at least exponentially distorted in mapping class groups with boundary components.
A closed hyperbolic Riemann surface M is said to be K-quasiconformally homogeneous if there exists a transitive family F of K-quasiconformal homeomorphisms. Further, if all [f] in F act trivially on H1(M;Z), we say M is Torelli-K-quasiconformally homogeneous. We prove the existence of a uniform lower bound on K for Tor…
We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are also considered. In addition, we prove that rational homology of the subgroup of t…
Study of Torelli groups of partitioned surfaces with bounds and asymptotic lengths.
problem Understanding Torelli groups of partitioned surfaces.
method Topological and dynamical analysis of Torelli groups of partitioned surfaces.
result Asymptotic translation lengths of Torelli groups of partitioned surfaces behave almost like the reciprocal of the Euler characteristic of the surface.
Theory connects Torelli subgroup homology to Sp(2g,ℤ)-modules.
problem Homology of Torelli subgroup of mapping class group.
method Equivariant group presentations and homology theory.
result Second homology group of Torelli subgroup is finitely generated as Sp(2g,ℤ)-module.
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.
New embedding connects Torelli group to skein module.
problem Understanding Torelli group structure.
method Embedding into Kauffman bracket skein algebra.
result New construction of Johnson homomorphism.
Study Heegaard Floer homology and word metric on Torelli group.
problem Relationship between Heegaard Floer homology and Torelli group word metric.
method Analyzes Heegaard Floer homology correction terms and word metric properties of Torelli group.
result Cayley graph of Torelli group has infinite diameter in word metric.
We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Joh…