An infinite presentation for the mapping class group of a nonorientable surface

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describes the linearity for the mapping class group of compact orientable surface with boundary, up to center.

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.

Omori and the author [R. Kobayashi and G. Omori, An infinite presentation for the mapping class group of a non-orientable surface with boundary, Osaka J. Math. 59(2) (2022) 269–314] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

Geometric intersection in representations of mapping class groups of surfaces

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, 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 construct the first known examples of nontrivial, normal, all pseudo-Anosov subgroups of mapping class groups of surfaces. Specifically, we construct such subgroups for the closed genus two surface and for the sphere with five or more punctures. Using the branched covering of the genus two surface over the sphere and results of Birman and Hilden, we prove that a reducible mapping class of the genus two surface projects to a reducible mapping class on the sphere with six punctures. The construction introduces "Brunnian" mapping classes of the sphere, which are analogous to Brunnian links.

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 prove that, for $$g\ge 19$$ the mapping class group of a nonorientable surface of genus g, $$\mathrm{Mod}(N_g)$$ , can be generated by two elements, one of which is of order g. We also prove that for $$g\ge 26$$ , $$\mathrm{Mod}(N_g)$$ can be generated by three involutions.

Let Mod(Sg) be the mapping class group of an orientable surface of genus g, Sg. The action of Mod(Sg) on the homology of Sg induces the well-known symplectic representation: Mod(Sg) ---> Sp(2g, Z). The kernel of this representation is called the Torelli group, I(Sg). We will study two subgroups of I(Sg). First we will look at the subgroup generated by all SIP-maps, SIP(Sg). We will show SIP(Sg) is not I(Sg) and is in fact an infinite index subgroup of I(Sg). We will also classify which SIP-maps are in the kernel of the Johnson homomorphism and Birman-Craggs-Johnson homomorphism. Then we will look at the symmetric Torelli group, SI(Sg). More specifically, we will investigate the group generated by Dehn twists about symmetric separating curves denoted H(Sg). We will show the well-known Birman-Craggs-Johnson homomorphism is not able to distinguish among SI(Sg), H(Sg), or K(Sg), where K(Sg) is the subgroup generated by Dehn twists about separating curves. Elements of H(Sg) act naturally on the symmetric separating curve complex, CH(S). We will show that when g > 4 Aut(CH(Sg)) = SMod^{+/-}(Sg) / < i > where SMod(Sg) is the symmetric mapping class group and i is a fixed hyperelliptic involution. Lastly we will give an algebraic characterization of Dehn twists about symmetric separating curves.

We compute the group of Morita self-equivalences (the Picard group) of a Poisson structure on an orientable surface, under the assumption that the degeneracies of the Poisson tensor are linear.The answer involves mapping class groups of surfaces, i.e., groups of isotopy classes of diffeomorphisms.We also show that the Picard group of these structures coincides with the group of outer Poisson automorphisms.

By classical results of Rochlin, Thom, Wallace and Lickorish, it is well-known that any two 3-manifolds (with diffeomorphic boundaries) are related one to the other by surgery operations. Yet, by restricting the type of the surgeries, one can define several families of non-trivial equivalence relations on the sets of (diffeomorphism classes of) 3-manifolds. In this expository paper, which is based on lectures given at the school “Winter Braids XI” (Dijon, December 2021), we explain how certain filtrations of mapping class groups of surfaces enter into the definitions and the mutual comparison of these surgery equivalence relations. We also survey the ways in which concrete invariants of 3-manifolds (such as finite-type invariants) can be used to characterize such relations.

A survey on cohomology of mapping class groups of surfaces and outer automorphism groups of free groups

Tits buildings $${\Delta }_{\mathbb{Q}}(\mathbf{G})$$ of linear algebraic groupsGdefined over the field of rational numbers $$\mathbb{Q}$$ have played an important role in understanding partial compactifications of symmetric spaces and compactifications of locally symmetric spaces, cohomological properties of arithmetic subgroups and S-arithmetic subgroups of $$\mathbf{G}(\mathbb{Q})$$ . Curve complexes $$\mathcal{C}({S}_{g,n})$$ of surfaces S g,n were introduced to parametrize boundary components of partial compactifications of Teichmüller spaces and were later applied to understand properties of mapping class groups of surfaces and geometry and topology of 3-dimensional manifolds. Tits buildings are spherical buildings, and another important class of buildings consists of Euclidean buildings, for example, the Bruhat–Tits buildings of linear algebraic groups defined over local fields. In this paper, we summarize and compare some properties and applications of buildings and curve complexes. We try to emphasize their similarities but also point out differences. In some sense, curve complexes are combinations of spherical, Euclidean and hyperbolic buildings. We hope that such a comparison might motivate more questions and also suggest methods to solve them at the same time, and furthermore it might introduce buildings to people who study curve complexes and curve complexes to people who study buildings.

These notes are the outcome of a mini-course on TQFTs held at the edition of Winter Braids in Pau in February 2015. We define the notion of TQFT and provide the first basic examples obtained via the universal construction and via Frobenius algebras. After recalling some basic notions on the mapping class groups of surfaces, we concentrate on the Reshetikhin-Turaev construction via the skein theoretical approach: we first define the skein module of a 3-manifold and the RT invariants; then we apply the universal construction to get the RT SU (2)-TQFTs. We conclude with an overview of the main results on these TQFTs and on some recent developments. An appendix summarizes the basic notions and facts in category theory used here.