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.
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…
Graph of groups palindromic width mostly infinite.
problem Understanding palindromic widths in graph of groups structures.
method Proved infinite palindromic width for specific graph of groups structures.
result Palindromic width of graph of groups mostly infinite.
Let F=<a,b> be a rank two free group. A word W(a,b) in F is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to a and b) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjuga…
The braid group Bn, endowed with Artin's presentation, admits two distinguished involutions. One is the anti-automorphism rev:Bn→Bn, v↦vˉ, defined by reading braids in the reverse order (from right to left instead of left to right). Another one is the conjugation $τ:x \mapsto Δ^{…
We give a unified geometric approach to some theorems about primitive elements and palindromes in free groups of rank 2. The geometric treatment gives new proofs of the theorems. Dedicated to Bill Harvey on his 65th birthday.
We consider non-elementary representations of two generator free groups in PSL(2,C), not necessarily discrete or free, G=<A,B>. A word in A and B, W(A,B), is a palindrome if it reads the same forwards and backwards. A word in a free group is {\sl primitive} if it is part of a minimal generating …
Introduces q-transpose for q-deformed modular group matrices.
problem Understanding q-deformed rational numbers and their properties. method Introduces q-transpose and applies it to refine q-deformed modular group actions. result New proof and refinement of Leclere and Morier-Genoud's trace palindromicity theorem.
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.
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.
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.
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.
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…
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…
For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli geometry of our surface and prove that every automorphism of the Torelli geometr…
New methods improve understanding of abelian subgroups in Torelli groups.
problem Understanding the rank of abelian subgroups in Torelli groups.
method New proofs of W. Vautaw's results, more transparent and stronger estimates.
result Stronger estimates of the rank of abelian subgroups with additional information.
Study of Torelli subgroups of automorphism groups of free groups.
problem Understanding the structure of Torelli subgroups of Out(Fn). method Adapting techniques from 3-manifolds and surface mapping class groups.
result These groups are finitely generated and satisfy a Birman exact sequence.
Paper proves Torelli group's finiteness for surfaces with 2 boundaries.
problem Finiteness of Torelli group for surfaces with 2 boundaries.
method Analyzes Torelli group properties for surfaces of genus ≥ 3 with 2 boundary components.
result Torelli group is finitely generated for surfaces of genus ≥ 3 with 2 boundaries.
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.
We prove that the cohomological dimension of the Torelli group for a closed connected orientable surface of genus g at least 2 is equal to 3g-5. This answers a question of Mess, who proved the lower bound and settled the case of g=2. We also find the cohomological dimension of the Johnson kernel (the subgroup of the To…
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.
Paper calculates Torelli group's cohomology second group.
problem Calculating the second rational cohomology group of the Torelli group.
method Building on Hain's and Kupers-Randal-Williams's work, the paper provides an exposition of prerequisite material and the two key results.
result Calculation of the second rational cohomology group of the Torelli group.
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. Enhances stability ranges for Torelli and congruence subgroup homologies.
problem Improving stability ranges for specific subgroup homologies.
method Analyzes H2(Torelli subgroup of Aut(Fn)'s), H2(Torelli subgroup of mapping class groups), and Hk(congruence subgroups of GL_n(R)'s).
result Improved central stability ranges for various subgroup homologies.
Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for …
In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The…
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.
Unified framework for studying Torelli group and congruence subgroup maps.
problem Understanding maps between Torelli group and congruence subgroup.
method Unified algorithms for computing Rochlin invariants and maps.
result Unified proofs and explicit evaluations of maps on Dehn twists.
New stability results for homology groups of Torelli subgroups and congruence subgroups.
problem Stability of homology groups of Torelli subgroups and congruence subgroups.
method Representation stability and syzygies of modules with finite polynomial degree.
result New stability results for the second homology groups of Torelli subgroups and homology of certain congruence subgroups.
Study of handlebody group intersections with Torelli and Johnson kernels.
problem Intersection of handlebody group with Torelli and Johnson kernels.
method Use Birman--Craggs--Johnson (BCJ) homomorphism to study intersections and compute cup products.
result Determine images of BCJ homomorphism restricted to handlebody group intersections and compute cup products.
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space called the handle graph on which the Torelli group acts cocompactly.
Graph cohomology solves symplectic problems in surface mapping groups.
problem Compute the symplectic decomposition of Torelli group and its cohomology.
method Graph cohomology and ideas from graph cohomology.
result Effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group.
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.
We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping…
The paper proves representation stability for Torelli groups and their filtrations.
problem Representation stability of Torelli groups and their filtrations.
method Uniform representation stability using rational VICQ-modules and SIQ-modules. result The quotients of the lower central series of Torelli subgroups are uniformly representation stable.
Torelli group cannot be realized as area-preserving homeomorphisms.
problem Realization of the Torelli group as area-preserving homeomorphisms.
method Analysis of the Torelli group and its relationship with homeomorphisms.
result The Torelli group has no realization inside the area-preserving homeomorphisms.
Minimal translation lengths for Torelli and pure braid groups on curve graphs are shown.
problem Understanding translation lengths of Torelli and pure braid groups on curve graphs.
method Analyzing asymptotic translation lengths of Torelli and pure braid groups on curve graphs.
result Minimal asymptotic translation lengths for Torelli and pure braid groups on curve graphs are shown to behave differently from their respective mapping class groups.
Proves a small set generates a subgroup of surface mapping classes.
problem Generating a subgroup of Torelli group efficiently.
method Proves genus one BP-maps generate handlebody subgroup.
result Proves a normal generating set for handlebody subgroup.
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.
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…
We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalga…