Mapping class group actions from Hopf monoids and ribbon graphs

Let Σ be an orientable surface of finite type and let Mod(Σ) be its mapping class group. We consider actions of Mod(Σ) by semisimple isometries on complete CAT(0) spaces. If the genus of Σ is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(Σ) on the completion of Teichmüller space with the Weil-Petersson metric shows that there are interesting actions of this type. Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it must fix a point. The mapping class group of a closed surface of genus 2 acts properly by semisimple isometries on a complete CAT(0) space of dimension 18.

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations $$\mathfrak{X} (M) = {\rm Hom} (\pi_1 (M), {\rm SU} (2)) / {\rm SU} (2).$$There is a natural action of the mapping class group of M on \({\mathfrak{X} (M)}\). We show here that this action is ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a map defined by the presentation of the surface group. This result is an extension of earlier results of Goldman for orientable surfaces (see [8]).

For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,\mu)$ is W$^*$-superrigid in the following sense: if $L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimes\Lambda$, for an arbitrary free ergodic pmp action $\Lambda\curvearrowright (Y,\nu)$, then the actions $\tilde B_n\curvearrowright X,\Lambda\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with a finite index subgroup of the direct product $\tilde B_{n_1}\times\cdots\times\tilde B_{n_k}$, for some $n_1,\ldots,n_k\geqslant 4$. The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from \cite{PV11} in combination with the OE superrigidity theorem for actions of mapping class groups from \cite{Ki06}.

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.

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 introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces S. They are derived from the parametrisation of the moduli spaces by the bundle of measured geodesic laminations over Teichm\"uller space of S and can be viewed as analytic continuations of the shear coordinates on Teichm\"uller space. In terms of these coordinates the gravitational symplectic structure takes a particularly simple form, which resembles the Weil-Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichm\"uller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichm\"uller space, corresponding to different values of the curvature.

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.

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

We consider the infinite dimensional Teichmuller space of a Riemann surface of general type. On the basis of the fact that the action of the quasiconformal mapping class group on the Teichmuller space is not discontinuous, in general, we divide the Teichmuller space into two disjoint subsets, the limit set and the region of discontinuity, according to the discreteness of the orbit by a subgroup of the quasiconformal mapping class group. The asymptotic Teichmuller space is a certain quotient space of the Teichmuller space and there is a natural projection from the Teichmuller space to the asymptotic Teichmuller space. We consider the fibers of the projection over any point in the asymptotic Teichmuller space, and show a coherence of the discreteness on each fiber in the Teichmuller space.

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth mathbb {Z}-graded gentle algebra A we associate a triple (Sigma _A, Lambda _A; eta _A), where Sigma _A is an oriented smooth surface with non-empty boundary, Lambda _A is a set of stops on partial Sigma _A and eta _A is a line field on Sigma _A, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Sigma _A, Lambda _A ;eta _A). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Sigma _A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichmuller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichmuller space, $T_{infty}$, was shown to carry on it a natural action of the universal commensurability mapping class group, $MC_{\infty}$. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surface. Utilizing a certain subgroup inside $MC_{\infty}$, (associated intimately to the inverse system of characteristic coverings), we can now descend to an inductive system of moduli spaces, and construct the direct limit ind-variety $M_{\infty}$. Invoking curvature properties of Quillen metrics on determinant bundles, and naturality under finite coverings of Weil-Petersson forms, we are able to construct on $M_{\infty}$ the natural sequence of determinant of cohomology line bundles, as well as the Mumford isomorphisms connecting these.

This self-contained paper is part of a series [FF2, FF3] on actions by diffeomorphisms of infinite groups on compact manifolds.The two main results presented here are:1. Any homomorphism of (almost any) mapping class group or automorphism group of a free group into Diff r + (S 1 ), r ≥ 2 is trivial.For r = 0 Nielsen showed that in many cases nontrivial (even faithful) representations exist.Somewhat weaker results are proven for finite index subgroups.2. We construct a finitely-presented group of real-analytic diffeomorphisms of R which is not residually finite.

We give two applications of our prior work toward the Putman–Wieland conjecture. First, we deduce a strengthening of a result of Marković–Tošić on virtual mapping class group actions on the homology of covers. Second, let and let be a finite ‐cover of topological surfaces. We show the virtual action of the mapping class group of on an ‐isotypic component of has nonunitary image.

Chapter 12 - Mapping Class Groups