Double Johnson filtrations for mapping class groups

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

We study decorated mapping class groups, i.e., mapping class groups of surfaces with marked points and boundary components, and their behaviour under stabilization maps with respect to the genus, the number of punctures and boundary components. Decorated mapping class groups are non-trivial extensions of the undecorated mapping class group, and the first result states that the extension is homologically trivial when one stabilizes with respect to the genus. The second result implies that one also gets splittings of homology groups when stabilizing with respect to the number of punctures and boundary components.

We prove that the Torelli group of an oriented 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 the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

Let Σg,1 be a compact oriented surface of genus g with one boundary component, and ℳg,1 its mapping class group. Morita showed that the image of the kth Johnson homomorphism τkℳ of ℳg,1 is contained in the kernel hg,1(k) of an Sp–equivariant surjective homomorphism H⊗ℤℒ2g(k+1)→ℒ2g(k+2), where H:=H1(Σg,1,ℤ) and ℒ2g(k) is the degree k part of the free Lie algebra ℒ2g generated by H. In this paper, we study the Sp–module structure of the cokernel hg,1ℚ(k)∕Im(τk,ℚℳ) of the rational Johnson homomorphism τk,ℚℳ:=τkℳ⊗ idℚ, where hg,1ℚ(k):=hg,1(k)⊗ℤℚ. In particular, we show that the irreducible Sp–module corresponding to a partition [1k] appears in the kth Johnson cokernel for any k≡1(mod4) and k≥5 with multiplicity one. We also give a new proof of the fact due to Morita that the irreducible Sp–module corresponding to a partition [k] appears in the Johnson cokernel with multiplicity one for odd k≥3. The strategy of the paper is to give explicit descriptions of maximal vectors with highest weight [1k] and [k] in the Johnson cokernel. Our construction is inspired by the Brauer–Schur–Weyl duality between Sp(2g,ℚ) and the Brauer algebras, and our previous work for the Johnson cokernel of the automorphism group of a free group.

We show that any pivotal Hopf monoid H in a symmetric monoidal category \mathcal{C} gives rise to actions of mapping class groups of oriented surfaces of genus g\geq 1 with n\geq 1 boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over H . They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where \mathcal{C} is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.

Given ϕ \phi a pseudo-Anosov map, let ℓ T ( ϕ ) \ell _\mathcal {T}(\phi ) denote the translation length of ϕ \phi in the Teichmüller space, and let ℓ C ( ϕ ) \ell _{\mathcal {C}}(\phi ) denote the stable translation length of ϕ \phi in the curve graph. Gadre–Hironaka–Kent–Leininger showed that, as a function of Euler characteristic χ ( S ) \chi (S) , the minimal possible ratio τ ( ϕ ) = ℓ T ( ϕ ) ℓ C ( ϕ ) \tau (\phi ) = \frac {\ell _\mathcal {T}(\phi )}{\ell _\mathcal {C}(\phi )} is log ⁡ ( | χ ( S ) | ) \log (|\chi (S)|) , up to uniform additive and multiplicative constants. In this short note, we introduce a new construction of such ratio optimizers and demonstrate their abundance in the mapping class group. Further, we show that ratio optimizers can be found arbitrarily deep into the Johnson filtration as well as in the point pushing subgroup.

Chapter 12 - Mapping Class Groups

It is shown that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups of surfaces with several boundary components.

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain lattices cannot have C^1 faithful actions on the circle. We also prove that for n > 5, any C^1 action of Aut(F_n) or Out(F_n) on the circle factors through an action of Z/2Z.

A linear representation of the mapping class group [formula omitted] and the theory of winding numbers

Let $\\Sigma_{g,1}$ be an oriented compact surface of genus $g$ with 1 boundary component, and $\\Gamma_{g,1}$ the mapping class group of $\\Sigma_{g,1}$. We determine the stable cohomology group of $\\Gamma_{g,1}$ with coefficients in $H^1 (\\Sigma_{g ,1} ; \\mathbb{Z})^{\\otimes n}$, $n \\ge 1$, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product $\\Gamma_{g,1} \\ltimes H_1 (\\Sigma_{g,1} ; \\mathbb{Z})$ (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes $\\widetilde{m_{i,j}}$'s $(i \\ge 0,\\, j \\ge 1,\\, i+j \\ge 2)$ [11] over the rational stable cohomology algebra of the group $\\Gamma_{g,1}$.

Finite presentations of the mapping class groups of once-stabilized Heegaard splittings DAIKI IGUCHILet g 2 and assume that we are given a genus g Heegaard splitting of a closed orientable 3-manifold with distance greater than 2g C 2. We prove that the mapping class group of the once-stabilization of such a Heegaard splitting is finitely presented.57K30, 57M60 1 Introduction Let .M; / be a Heegaard splitting of a compact orientable 3-manifold M .The mapping class group MCG.M; / of the Heegaard splitting .M; / is defined to be the group 0 .Diff.M; // of path-connected components of the group Diff.M; /, where we denote by Diff.M; / the group of diffeomorphisms of M that preserve setwise.There is a natural homomorphism from MCG.M; / to the mapping class group MCG.M / of M .Following Johnson [2011], we call the kernel of this natural homomorphism the isotopy subgroup of MCG.M; /, and denote it by Isot.M; /.In this paper, we are interested in the isotopy subgroup of the mapping class group of a once-stabilized Heegaard splitting.Let .M; 0 / be a genus g. 0 / 2 Heegaard splitting of a closed orientable 3manifold M .We say that a Heegaard splitting .M; / is a (once-)stabilization of .M; 0 / if it is obtained from .M; 0 / by adding a 1-handle whose core is parallel into 0 .Corresponding to two handlebodies V 0 and V C 0 in M with @V 0 D @V C 0 D 0 , there are two obvious subgroups of Isot.M; /: one is Isot.V 0 ; / and the other is Isot.V C 0 ; C /, where (resp. C ) is the Heegaard surface obtained by pushing into V 0 (resp.V C 0 ) slightly.It is natural to ask when these subgroups generate Isot.M; /.Johnson [2011] proved that if the distance d. 0 / of the Heegaard splitting .M; 0 / is greater than 2g. 0 / C 2, then the two subgroups defined above generate Isot.M; /.As a consequence of this fact, together with a result of Scharlemann [2013] that says Isot.V 0 ; / are finitely generated, it follows that Isot.M; / and MCG.M; / are finitely generated.In that paper, Johnson conjectured that Isot.M; / is an amalgamation of the two groups Isot.V 0 ; / and Isot.V C 0 ; C /.This is the main result of the paper:Theorem 1.1 Suppose that .M; 0 / is Heegaard splitting of a closed orientable 3-manifold M with d. 0 / > 2g. 0 / C 2, and that .M; / is a once-stabilization of .M; 0 /.Suppose that .V 0 ; / (resp.

Formanek and Procesi have demonstrated that Aut(F_n) is not linear for n >2.\nTheir technique is to construct nonlinear groups of a special form, which we\ncall FP-groups, and then to embed a special type of automorphism group, which\nwe call a poison group, in Aut(F_n), from which they build an FP-group. We\nfirst prove that poison groups cannot be embedded in certain mapping class\ngroups. We then show that no FP-groups of any form can be embedded in mapping\nclass groups. Thus the methods of Formanek and Procesi fail in the case of\nmapping class groups, providing strong evidence that mapping class groups may\nin fact be linear.\n

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, namely the general linear groups ${\rm{GL}}(n,\Z)$, the groups of outer automorphisms of free groups of rank $n\geq 2$, and the mapping class groups of closed orientable surfaces of genus $g\geq 1$. Much of the work on mapping class groups and automorphisms of free groups is motivated by the idea that these sequences of groups are strongly analogous. In this article we highlight a few of the most striking similarities and differences between these series of groups and present a list of open problems motivated by this philosophy.

We refine the construction of quasi-homomorphisms on mapping class groups. It is useful to know that there are unbounded quasi-homomorphisms which are bounded when restricted to particular subgroups since then one deduces that the mapping class group is not boundedly generated by these subgroups. In this note we enlarge the class of such subgroups. The generalization is motivated by considerations in first order theory of free groups.