arXiv research

Generators found for a specific subgroup of braid groups.

In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was previously announced by Benson Farb for genus at least 4 and has recently been announc…

In this paper, we show that the minimal asymptotic translation length of the Torelli group $\mathcal{I}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group $Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translat…

In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical …

Symplectic Torelli groups of positive rational surfaces are trivial or sphere braid groups.

We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. As one application, we establish the automorphism groups of each t…

Study on embedding tree products into groups, distinguishing them.

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…

Improved lower bounds for faithful linear representations of mapping class groups.

We investigate the cohomology of the level 4 subgroup of the braid group, namely, the kernel of the mod 4 reduction of the Burau representation at $t=-1$. This group is also equal to the kernel of the mod 2 abelianization of the pure braid group. We give an exact formula for the first Betti number; it is a quartic poly…

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute low…

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 …

We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polyn…

In the study of the relation between the mapping class group M of a surface and the theory of finite-type invariants of homology 3-spheres, three subgroups of the mapping class group play a large role. They are the Torelli group, the Johnson subgroup K and a new subgroup L, which contains K, defined by a choice of a La…

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, w…

Researchers found the second homology of Torelli groups for large g.

Torelli group's second homology is finite for large genus surfaces.

Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating…

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.

Study homology groups of mapping and Torelli groups for surfaces with abelian covers.

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…

Study Heegaard Floer homology and word metric on Torelli group.

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…

Study of Torelli subgroups of automorphism groups of free groups.

Paper proves Torelli group's finiteness for surfaces with 2 boundaries.

Proves properties of Torelli Lie algebra for surfaces.

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…

Study Torelli subgroups of handlebody groups and their cohomology.

The contraction of the image of the Johnson homomorphism is called the Chillingworth class. In this paper, we derive a combinatorial description of the Chillingworth class for Putman's subsurface Torelli groups. We also prove the naturality and uniqueness properties of the map whose image is the dual of the Chillingwor…

Paper calculates Torelli group's cohomology second group.

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class gro…

Enhances stability ranges for Torelli and congruence 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…

Unified framework for studying Torelli group and congruence subgroup maps.

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The present paper is devoted to a new approach to the results of W. Vautaw about Dehn multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new proofs are more tran…

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.

We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that o…

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…

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 study finds all trace field degrees for Torelli group mappings.

Symplectic manifolds derived from Torelli groups, showing complex structure invariants without holomorphic structures.

We develop an analogue of the Birman exact sequence for the Torelli subgroup of Aut(F_n). This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group Aut(F_n). These results play an important role in the authors' recent work on the second homology group of the To…

The aim of this article is to prove that the Torelli group action on the G-character varieties is ergodic for G a connected, semi-simple and compact Lie group.

For all but finitely many compact orientable surfaces, we show that any superinjective map from the complex of separating curves into itself is induced by an element of the extended mapping class group. We apply this result to proving that any finite index subgroup of the Johnson kernel is co-Hopfian. Analogous propert…

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact connected subsurface of a closed surface can be extended to a diffeomorphism of t…