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 polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3-spheres.
We observe that the determinant of the representation provides a little restriction for the structure of the graded quotients $\mathcal{G}_k^Q{F}$ introduced in both [Algebr. Geom. Topol. 1 (2001) 39-55] and [J. Knot Theory Ramifications 13 (2004) 297-306] that any one of them does not contain the trivial 1-dimensional summand.
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided g > 3), and (3) infinitely many genus g Lefschetz fibrations over genus h surfaces that are not fiber sums of holomorphic ones.
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 S into itself, and show that any finite index subgroup of the Torelli group for S is co-Hopfian.
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 announced by Benson Farb and Nikolai Ivanov for subgroups of finite index in the Torelli group for surfaces of genus at least 5. This result is also directly analogous to previous results established for classical braid groups, mapping class groups, and surface braid groups.
Torelli space (in genus g) is the moduli space of compact Riemann surfaces of genus g together with a symplectic basis of their first homology group. It is the quotient of the genus g Teichmuller space by the Torelli group T_g and is a model of the classifying space of T_g. It is known that almost all T_g are not finite complexes. The goal of this note is to present a suite of problems designed to probe the infinite topology of Torelli spaces. Some background is given and a few new results are proved.
Genus $g$ Torelli space is the moduli space of genus $g$ curves of compact type equipped with a homology framing. The hyperelliptic locus is a closed analytic subvariety consisting of finitely many mutually isomorphic components. We use properties of the hyperelliptic Torelli group to show that when $g\geq 3$ these components do not have the homotopy type of a finite CW complex. Specifically, we show that the second rational homology of each component is infinite-dimensional. We give a more detailed description of the topological features of these components when $g=3$ using properties of genu…
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 Torelli group generated by Dehn twists about separating curves) to be 2g-3. For g at least 2, we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli…
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 the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera > 3; they are quadratic. We also show th…
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.
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.
For a closed surface $S$, its Torelli group $\mathcal{I}(S)$ is the subgroup of the mapping class group of $S$ consisting of elements acting trivially on $H_1(S;\mathbb{Z})$. When $S$ is orientable, a generating set for $\mathcal{I}(S)$ is known. In this paper, we give a normal generating set of $\mathcal{I}(N_g)$ for $g\geq4$, where $N_g$ is a genus-$g$ non-orientable closed surface.
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 category $\TSur$ whose objects are surfaces together with a decoration restricting how they can be embedded into larger surfaces and whose morphisms are embeddings which respect the decoration. There is a natural ``Torelli functor'' on this category which extends the usual definition of…
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 main result of this paper shows that Torelli buildings of surfaces of genus at least 5 have only obvious automorphisms, and identifies its group of automorphisms. Namely, we prove that for such a surface every automorphism of its Torelli building is induced by a diffeomorphism o…
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 curves within a fixed homology class is connected.
Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.
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 geometry is induced by a diffeomorphism of the surface in question. We also provide an intrinsic algebraic characterization of some natural elements of the Torelli group and of some geometric relations between them. When combined, these results allow us to compute the automorphism group…
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 generating set for $\mathcal{I}(N_g^b)$, where $N_g^b$ is a genus-$g$ compact non-orientable surface with $b$ boundary components for $g\geq4$ and $b\geq1$.
Let S be a closed oriented surface of genus g > 1, and let T denote its Torelli group. First, given a set E of homotopically nontrivial, pairwise disjoint, pairwise nonisotopic simple closed curves on S, we determine precisely when a multitwist on E is an element of T by defining an equivalence relation on E and then applying graph theory. Second, we prove that an arbitrary Abelian subgroup of T has rank < 2g-4.
We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least four. We also prove that, in this range, the $I$-adic completion of the Alexander invariant is finite-dimensional, and the Kahler property for the Torelli group implies t…
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 of the mapping group with coefficients in a symplectic representation $V$. Using ideas from graph cohomology, we give an effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group, and assuming the later is Kozsul…
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 translation length of the pure braid group $PB_n$ on the curve graph asymptotically behaves like $1/n$, contrary to the braid group $B_n$, which behaves like $1/n^2$.
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 g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dim…
For an oriented virtual link, L.H. Kauffman defined the f-polynomial (Jones polynomial). The supporting genus of a virtual link diagram is the minimal genus of a surface in which the diagram can be embedded. In this paper we show that the span of the f-polynomial of an alternating virtual link L is determined by the number of crossings of any alternating diagram of L and the supporting genus of the diagram. It is a generalization of Kauffman-Murasugi-Thistlethwaite's theorem. We also prove a similar result for a virtual link diagram that is obtained from an alternating virtual link diagram by …
Let I_g,* denote the (pointed) Torelli group. This is the group of homotopy classes of homeomorphisms of the genus g >= 2 surface S_g with a marked point, acting trivially on H := H_1(S_g). In 1983 Johnson constructed a beautiful family of invariants tau_i: H_i(I_g,*) -> /\^{i+2} H for 0 <= i <= 2g-2, using a kind of Abel-Jacobi map for families, in order to detect nontrivial cycles in I_g,*. Johnson proved that tau_1 is an isomorphism rationally, and asked if the same is true for tau_i with i > 1. The goal of this paper is to introduce various methods for computing tau_i; in particular we pro…
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 show that the kernel of the Magnus representation, Mag(S), is highly non-trivial and has a rich structure as a group. Specifically, we define an infinite filtration of Mag(S) by subgroups, called the higher order Magnus subgroups, M_k(S). We develop methods for generating nont…
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block.
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(F_n). The main tools used in the proofs are the related theories of FI-modules (due to the first author together…
It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1. We extend this result to almost positive links and partly identify the 3 following coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property of canonical Seifert surfaces associated to almost positive (and almost alternating) link diagrams.
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 corresponding mapping class group when the genus is at least 3. As an application, we prove that when g > 2, the action of the pro-ell completion of the Torelli group T_{g,1} on the pro-ell fundamental group of a pointed genus g surface is not faithful. The choice of a first-order…
We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.
We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.
For each discriminant $D>1$, McMullen constructed the Prym-Teichmüller curves $W_D(4)$ and $W_D(6)$ in $\mathcal{M}_{3}$ and $\mathcal{M}_{4}$, which constitute one of the few known infinite families of geometrically primitive Teichmüller curves. In the present paper, we determine for each $D$ the number and type of orbifold points on $W_D(6)$. These results, together with a previous result of the two authors in the genus $3$ case and with results of Lanneau-Nguyen and Möller, complete the topological characterisation of all Prym-Teichmüller curves and determine their genus. The study of orbif…
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot…
Let $Γ$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Γ$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Γ;\Q) \cong \Q$ for $g \geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\Q$. We also prove analogous results for surface with punctures and boundary components.
Let $Γ$ be either the mapping class group of a closed surface of genus $\geq 2$, or the automorphism group of a free group of rank $\geq 3$. Given any homological representation $ρ$ of $Γ$ corresponding to a finite cover, and any term $\mathcal{I}_k$ of the Johnson filtration, we show that $ρ(\mathcal{I}_k)$ has finite index in $ρ(\mathcal{I})$, the Torelli subgroup of $Γ$. Since $[\mathcal{I}: \mathcal{I}_k] = \infty$ for $k > 1$, this implies for instance that no such representation is faithful.
Recently, Gareth Jones observed that every finite group $G$ can be realized as the group of automorphisms of some dessin d'enfant ${\mathcal D}$. In this paper, complementing Gareth's result, we prove that for every possible action of $G$ as a group of orientation-preserving homeomorphisms on a closed orientable surface of genus $g \geq 2$, there is a dessin d'enfant ${\mathcal D}$ admitting $G$ as its group of automorphisms and realizing the given topological action. In particular, this asserts that the strong symmetric genus of $G$ is also the minimum genus action for it to acts as the group…
For any positive integer $n$, we exhibit a cofinite subgroup $Γ_n$ of the mapping class group of a surface of genus at most two such that $Γ_n$ admits an epimorphism onto a free group of rank $n$. We conclude that $H^1(Γ_n;\Z)$ has rank at least $n$ and the dimension of the second bounded cohomology of each of these mapping class groups is the cardinality of the continuum. In the case of genus two, the groups $Γ_n$ can be chosen not to contain the Torelli group. Similarly for hyperelliptic mapping class groups. We also exhibit an automorphism of a subgroup of finite index in the mapping class …
The rational cohomology ring of A_3, the moduli space of abelian 3-folds is computed. This is isomorphic to the the rational cohomology ring of the group Sp_3(Z) of 6x6 integral symplectic matrices. The main ingredients in the computation are (1) Looijenga's computation of the rational cohomology ring of M_3, the moduli space of smooth projective curves of genus 3, and (2) the stratified Morse theory of Goresky and MacPherson, which we use to compute the homology of the jacobian locus (in the rank 3 Siegel upper half plane), or equivalently of the extended Torelli group in genus 3. In the revi…
Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to ${1/2} \wedge^3 H \semi \sp(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in ${1/2} \wedge^3 H \semi \sp(H)$. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebo…
Let R be a compact oriented surface of genus g with one boundary component. Homology cylinders over R form a monoid IC into which the Torelli group I of R embeds by the mapping cylinder construction. Two homology cylinders M and M' are said to be Y_k-equivalent if M' is obtained from M by "twisting" an arbitrary surface S in M with a homeomorphim belonging to the k-th term of the lower central series of the Torelli group of S. The J_k-equivalence relation on IC is defined in a similar way using the k-th term of the Johnson filtration. In this paper, we characterize the Y_3-equivalence with thr…
We show that no finite index subgroup of a sufficiently complicated mapping class group or braid group can act faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on the circle, which generalizes a result of Farb-Franks, and which parallels a result of Ghys and Burger-Monod concerning differentiable actions of higher rank lattices on the circle. This answers a question of Farb, which has its roots in the work of Nielsen. We prove this result by showing that if a right-angled Artin group acts faithfully by $C^{1+\mathrm{bv}}$ diffeomorphisms on a compact one-manifold, then its defining graph has …
