Circularly ordering direct products and the obstruction to left-orderability

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.

Let arc A A in the conformal plane or on the sphere have local cyclic order three and cyclic order t t . It can be decomposed into a finite number of subarcs of cyclic order three. Let the disection order of A A be the minimum number of arcs in such a decomposition. The principal result of this paper is that the cyclic order t t and dissection order d d of A A satisfy the relation d + 2 ≦ t ≦ 3 d d + 2 \leqq t \leqq 3d . In establishing this result it is proved that a necessary and sufficient condition, that an arc of local cyclic order three shall be of global cyclic order three, is that there exists a circle meeting it only at the endpoints.

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.

Geometric intersection in representations of mapping class groups of surfaces

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.

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.

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.

On some operations and density of m-polar fuzzy graphs

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.

Existence of BSAs and cyclic BSAs of block size three

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.

We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.

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.

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.