Chapter 12 - 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

Homological representations of low genus mapping class groups

Let M n M_n be the connect sum of n n copies of S 2 × S 1 S^2 \times S^1 . A classical theorem of Laudenbach says that the mapping class group Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is an extension of Out ⁡ ( F n ) \operatorname {Out}(F_n) by a group ( Z / 2 ) n (\mathbb {Z}/2)^n generated by sphere twists. We prove that this extension splits, so Mod ⁡ ( M n ) \operatorname {Mod}(M_n) is the semidirect product of Out ⁡ ( F n ) \operatorname {Out}(F_n) by ( Z / 2 ) n (\mathbb {Z}/2)^n , which Out ⁡ ( F n ) \operatorname {Out}(F_n) acts on via the dual of the natural surjection Out ⁡ ( F n ) → G L n ( Z / 2 ) \operatorname {Out}(F_n) \rightarrow GL_n(\mathbb {Z}/2) . Our splitting takes Out ⁡ ( F n ) \operatorname {Out}(F_n) to the subgroup of Mod ⁡ ( M n ) \operatorname {Mod}(M_n) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M n M_n . Our techniques also simplify various aspects of Laudenbach’s original proof, including the identification of the twist subgroup with ( Z / 2 ) n (\mathbb {Z}/2)^n .

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

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.

In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier [6]. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any “sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous flip graphs of infinite-type surfaces.

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.

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.

We rene 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 rst order theory of free groups.

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product of this investigation, we determine the second homology of the mapping class group of a surface of genus 3.

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.

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 first give an exposition of how the Polyakov path integral for the bosonic string produces a natural mapping class group invariant measure, $d(Poly)$, on the Teichmuller space of Riemann surfaces of each fixed genus. The description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles over Teichmuller space is also explained. We then report on our recent results with I.Biswas and D.Sullivan, where we succeed in coherently fitting together this Polyakov-Mumford construction over the universal direct limit, $T_\infty$, of all the finite genus Teichmuller spaces. The virtual automorphism group of the fundamental group of a compact surface (of arbitrary genus $g > 1$) acts by automorphisms on $T_\infty$ as the ``commensurability mapping class group''. Our entire construction is equivariant with respect to this new modular group.

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.

Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstrating that the algebraic structure of $\mathrm{PMap}(S)$ can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that $\mathrm{Map}(S)$ fails to be residually finite for any infinite-type surface $S$. In addition, we give a topological generating set for $\mathrm{PMap}(S)$ equipped with the compact-open topology. In particular, if $S$ has at most one end accumulated by genus, then $\mathrm{PMap}(S)$ is topologically generated by Dehn twists, otherwise the Dehn twists along with handle shifts topologically generate.