Semisimple actions of mapping class groups on CAT(0) spaces

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.

The study of the mapping class group Mod( S ) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod( S ), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod( S ) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

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 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

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]).

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.

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}.

Chapter 12 - Mapping Class Groups

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.

In this paper, we study isometric actions on finite-dimensional CAT(0) spaces for the Higman–Thompson groups T_n, which are generalizations of Thompson’s group T. It is known that every semi-simple action of T on a complete CAT(0) space of finite covering dimension has a global fixed point. After this result, we show that every semi-simple action of T_n on a complete CAT(0) space of finite covering dimension has a global fixed point. In the proof, we regard T_n as ring groups of homeomorphisms of S^1 introduced by Kim, Koberda and Lodha, and use general facts on these groups.

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.

Formanek and Procesi have demonstrated that Aut(F_n) is not linear for n &gt;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

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 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.

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.