arXiv research

Study genus-three Torelli maps and their fixed point sets in representation varieties.

Unified framework for studying Torelli group and congruence subgroup maps.

Enhances stability ranges for Torelli and congruence subgroup homologies.

Study Heegaard Floer homology and word metric on Torelli group.

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

The study finds all trace field degrees for Torelli group mappings.

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…

Study of Torelli subgroups of automorphism groups of free groups.

We show that for all but finitely many compact orientable surfaces, any superinjective map from the complex of separating curves into the Torelli complex is induced by an element of the extended mapping class group. As an application, we prove that any injective homomorphism from a finite index subgroup of the Johnson …

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 …

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…

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…

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

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…

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…

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.

In this note we survey recent results on the extrinsic geometry of the Jacobian locus inside $\mathsf{A}_g$. We describe the second fundamental form of the Torelli map as a multiplication map, recall the relation between totally geodesic subvarieties and Hodge loci and survey various results related to totally geodesic…

New descriptions of a subgroup in mapping class groups.

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 Bergman kernel and period map for curves are studied.

We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $π_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give counterexamples to Theorem 3.4 of Verbitsky's paper "Mapping class group and a global …

A mapping class group of an oriented manifold is a quotient of its diffeomorphism group by the isotopies. We compute a mapping class group of a hypekahler manifold $M$, showing that it is commensurable to an arithmetic subgroup in SO(3, b_2-3). A Teichmuller space of $M$ is a space of complex structures on $M$ up to is…

The mapping class group of a closed surface of genus $g$ is an extension of the Torelli group by the symplectic group. This leads to two natural problems: (a) compute (stably) the symplectic decomposition of the lower central series of the Torelli group and (b) compute (stably) the Poincaré polynomial of the cohomology…

The paper proves unique characterization of gravitational instantons with specific volume growth.

We give a new proof of the theorem of Birman-Powell that the Torelli subgroup of the mapping class group of a closed orientable surface of genus at least 3 is generated by simple homeomorphisms known as bounding pair maps. The key ingredient is a proof that the subcomplex of the curve complex of the surface spanned by …

Study on torsion in homology of Torelli group for surfaces.

For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of $…

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…

Study abelian cycles in Torelli group homology, proving new results in stable rational homology.

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…

Study Torelli subgroups of handlebody groups and their cohomology.

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all "separating twists", all "bounding pair maps", and all "commutators of simply intersecting pairs" and whose relations all come from a short list of topological …

For a compact surface $S$, let $\mathcal{I}(S)$ denote the Torelli group of $S$. For a compact orientable surface $Σ$, $\mathcal{I}(Σ)$ is generated by BSCC maps and BP maps. For a non-orientable closed surface $N$, $\mathcal{I}(N)$ is generated by BSCC maps and BP maps. In this paper, we give an explicit normal genera…

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 develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,\mathbb{Z})$-module.

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.

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…

Let {a,b} and {c,d} be two pairs of bounding simple closed curves on an oriented surface which intersect nontrivialy. We prove that if these pairs are invariant under the action of an orientation reversing involution, then the corresponding bounding pair maps generate a free group. This supports the conjecture stated b…

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 …

Compact hyperkaehler manifolds are higher-dimensional generalizations of K3 surfaces. The classical Global Torelli theorem for K3 surfaces, however, does not hold in higher dimensions. More precisely, a compact hyperkaehler manifold is in general not determined by its natural weight-two Hodge structure. The text gives …

We study the complex of partial bases of a free group, which is an analogue for $\Aut(F_n)$ of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of $\Aut(F_n)$ is highly connected. Using these results, we give a…

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…

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class…

Study subgroup actions on mapping class groups using Heisenberg representations.

The paper calculates the top homology group of a specific Torelli group.

Study shows compact mapping class groups of infinite type surfaces are never perfect.

Boundary Dehn twist on $K3$ surfaces becomes trivial after abelianization.