Mapping Class Group Research Articles

A projection from filling currents to Teichmüller space

Let S S be a closed, genus g g surface. The space of geodesic currents on S S encompasses the set of closed curves up to homotopy, as well as Teichmüller space, and many other spaces of structures on S S . We show that one can define a mapping class group equivariant, length minimizing projection from the set of filling geodesic currents down to Teichmüller space, and prove some basic properties of this projection to show that it is well-behaved.

The differential graded Verlinde formula and the Deligne Conjecture

AbstractA modular category gives rise to a differential graded modular functor, that is, a system of projective mapping class group representations on chain complexes. This differential graded modular functor assigns to the torus the Hochschild chain complex and, in the dual description, the Hochschild cochain complex of . On both complexes, the monoidal product of induces the structure of an ‐algebra, to which we refer as the differential graded Verlinde algebra. At the same time, the modified trace induces on the tensor ideal of projective objects in a Calabi–Yau structure so that the cyclic Deligne Conjecture endows the Hochschild cochain and chain complex of with a second ‐structure. Our main result is that the action of a specific element in the mapping class group of the torus transforms the differential graded Verlinde algebra into this second ‐structure afforded by the Deligne Conjecture. This result is established for both the Hochschild chain and the Hochschild cochain complex of . In general, these two versions of the result are inequivalent. In the case of Hochschild chains, we obtain a block diagonalization of the Verlinde algebra through the action of the mapping class group element . In the semisimple case, both results reduce to the Verlinde formula. In the non‐semisimple case, we recover after restriction to zeroth (co)homology earlier proposals for non‐semisimple generalizations of the Verlinde formula.

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

The handlebody group and the images of the second Johnson homomorphism

Given an oriented surface bounding a handlebody, we study the subgroup of its mapping class group defined as the intersection of the handlebody group and the second term of the Johnson filtration: $\mathcal{A} \cap J_2$. We introduce two trace-like operators, inspired by Morita's trace, and show that their kernels coincide with the images by the second Johnson homomorphism $\tau_2$ of $J_2$ and $\mathcal{A} \cap J_2$, respectively. In particular, we answer by the negative to a question asked by Levine about an algebraic description of $\tau_2(\mathcal{A} \cap J_2)$. By the same techniques, and for a Heegaard surface in $S^3$, we also compute the image by $\tau_2$ of the intersection of the Goeritz group $\mathcal{G}$ with $J_2$.

Counting arcs on hyperbolic surfaces

We give the asymptotic growth of the number of arcs of bounded length between boundary components on hyperbolic surfaces with boundary. Specifically, if S has genus g , n boundary components and p punctures, then the number of orthogeodesic arcs in each pure mapping class group orbit of length at most L is asymptotic to L^{6g-6+2(n+p)} times a constant. We prove an analogous result for arcs between cusps, where we define the length of such an arc to be the length of the sub-arc obtained by removing certain cuspidal regions from the surface.

Moduli spaces of complex affine and dilation surfaces

Abstract We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ⁢ ( 𝒎 ) {{\mathcal{A}}_{g,n}(\boldsymbol{m})} of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 ⁢ … , m n ) {\boldsymbol{m}=(m_{1}\ldots,m_{n})} is a holomorphic affine bundle over ℳ g , n {\mathcal{M}_{g,n}} , and the moduli space 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} of dilation surfaces is a covering space of ℳ g , n {\mathcal{M}_{g,n}} . We then classify the connected components of 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} and show that it is an orbifold- K ⁢ ( G , 1 ) {K(G,1)} , where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].

Mapping classes are almost determined by their finite quotient actions

Given any connected compact orientable surface, a pair of mapping classes is said to be procongruently conjugate if it induces a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any pseudo-Anosov mapping class of a connected closed orientable surface, several topological features are confirmed to depend only on the procongurent conjugacy class of the mapping class, including the stretching factor, the topological type of the prong singularities, the transverse orientability of the invariant foliations, and the isomorphism type of the symplectic Floer homology.

Stable cohomology of graph complexes

We study three graph complexes related to the higher genus Grothendieck–Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus g tends to infty . As a byproduct, we find that the Malcev completion of the genus g mapping class group relative to the symplectic group is Koszul in the stable limit, partially answering a question of Hain.

The mapping class group of connect sums of 𝑆²×𝑆¹

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 .

Cohomology of configuration spaces of surfaces as mapping class group representations

We express the rational cohomology of the unordered configuration space of a compact oriented manifold as a representation of its mapping class group in terms of a weight-decomposition of the rational cohomology of the mapping space from the manifold to a sphere. We apply this to the case of a compact oriented surface with one boundary component and explicitly compute the rational cohomology of its unordered configuration space as a representation of its mapping class group. In particular, this representation is not symplectic, but has trivial action of the second Johnson filtration subgroup of the mapping class group.

Coarse geometry of pure mapping class groups of infinite graphs

We discuss the large-scale geometry of pure mapping class groups of locally finite, infinite graphs, motivated by recent work of Algom-Kfir–Bestvina [1] and the work of Mann–Rafi [19] on the large-scale geometry of mapping class groups of infinite-type surfaces. Using the framework of Rosendal [24] for coarse geometry of non-locally compact groups, we classify when the pure mapping class group of a locally finite, infinite graph is globally coarsely bounded (an analog of compact) and when it is locally coarsely bounded (an analog of locally compact).Our techniques give lower bounds on the first integral cohomology of the pure mapping class group for some graphs and allow us to compute the asymptotic dimension of all locally coarsely bounded pure mapping class groups of infinite rank graphs. This dimension is always either zero or infinite.

Factoring periodic maps into Dehn twists

Let Mod ( S g ) be the mapping class group of the closed orientable surface S g of genus g ≥ 1 . In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order 4 g and 4 g + 2 , for g ≥ 2 , whose product is pseudo-Anosov.

The limit set of non-orientable mapping class groups

The limit set of non-orientable mapping class groups

Alternative versions of the Johnson homomorphisms and the LMO functor

Let $\Sigma$ be a compact connected oriented surface with one boundary component and let $\mathcal{M}$ denote the mapping class group of $\Sigma$. By considering the action of $\mathcal{M}$ on the fundamental group of $\Sigma$ it is possible to define different filtrations of $\mathcal{M}$ together with some homomorphisms on each term of the filtration. The aim of this paper is twofold. Firstly we study a filtration of $\mathcal{M}$ introduced recently by Habiro and Massuyeau, whose definition involves a handlebody bounded by $\Sigma$. We shall call it the "alternative Johnson filtration", and the corresponding homomorphisms are referred to as "alternative Johnson homomorphisms". We provide a comparison between the alternative Johnson filtration and two previously known filtrations: the original Johnson filtration and the Johnson-Levine filtration. Secondly, we study the relationship between the alternative Johnson homomorphisms and the functorial extension of the Le-Murakami-Ohtsuki invariant of $3$-manifolds. We prove that these homomorphisms can be read in the tree reduction of the LMO functor. In particular, this provides a new reading grid for the tree reduction of the LMO functor.

Matrix group integrals, surfaces, and mapping class groups II: $$\textrm{O}\left( n\right) $$ and $$\textrm{Sp}\left( n\right) $$

Let w be a word in the free group on r generators. The expected value of the trace of the word in r independent Haar elements of \(\textrm{O}(n)\) gives a function \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) of n. We show that \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) has a convergent Laurent expansion at \(n=\infty \) involving maps on surfaces and \(L^{2}\)-Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning \({\textrm{U}}\left( n\right) \), yet with some surprising twists. A priori to our result, \({\mathcal {T}}r_{w} ^{\textrm{O}}(n)\) does not change if w is replaced with \(\alpha (w)\) where \(\alpha \) is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under \(\textrm{Aut}({\textbf{F}} _{r})\). As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of \({\mathcal {T}}r _{w}^{\textrm{O}}(n)\) as \(n\rightarrow \infty \), we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani. Our results are obtained more generally for a tuple of words \(w_{1},\ldots ,w_{\ell }\), leading to functions \({\mathcal {T}}r_{w_{1},\ldots ,w_{\ell }} ^{\textrm{O}}\). We also obtain all the analogous results for the compact symplectic groups \(\textrm{Sp}(n)\) through a rather mysterious duality formula.

From braid groups to mapping class groups

In this paper, we classify homomorphisms from the braid group on n strands to the mapping class group of a genus g surface. In particular, we show that when \(g<n-2\), all representations are either cyclic or standard. Our result is sharp in the sense that when \(g=n-2\), a generalization of the hyperelliptic representation appears, which is not cyclic or standard. This gives a classification of surface bundles over the configuration space of the complex plane. As a corollary, we partially recover the result of Aramayona–Souto (Geom Topol 16(4):2285–2341, 2012), which classifies homomorphisms between mapping class groups, with a slight improvement.

Twisted conjugacy in big mapping class groups

Let G be a group and φ be an automorphism of G. Two elements x,y of G are said to be φ-twisted conjugate if y=gxφ(g)−1 for some g∈G. A group G has the R∞-property if the number of φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this paper, we prove that the big mapping class group MCG(S) possesses the R∞-property under some suitable conditions on the infinite-type surface S. As an application, we also prove that the big mapping class group possesses the R∞-property if and only if it satisfies the S∞-property.

Arithmetic quotients of the automorphism group of a right-angled Artin group

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, \operatorname{Aut}(F_n) , has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of \operatorname{Aut}(F_n) for n \geq 4 where k th powers of all transvections act trivially for some fixed k . Thus, for some values of k , we deduce that the quotient of \operatorname{Aut}(F_n) by the subgroup generated by k th powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann.

Extending periodic maps on surfaces over the 4-sphere

Let [Formula: see text] be the closed orientable surface of genus [Formula: see text]. We address the problem to extend torsion elements of the mapping class group [Formula: see text] over the 4-sphere [Formula: see text]. Let [Formula: see text] be a torsion element of maximum order in [Formula: see text]. Results including: (1) For each [Formula: see text], [Formula: see text] is periodically extendable over [Formula: see text] for some non-smooth embedding [Formula: see text], and not periodically extendable over [Formula: see text] for any smooth embedding [Formula: see text]. (2) For each [Formula: see text], [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if and only if [Formula: see text]. (3) Each torsion element of order [Formula: see text] in [Formula: see text] is extendable over [Formula: see text] for some smooth embedding [Formula: see text] if either (i) [Formula: see text] and [Formula: see text] is even; or (ii) [Formula: see text] and [Formula: see text]; or (iii) [Formula: see text]. Moreover, the conditions on [Formula: see text] in (i) and (ii) cannot be removed.

A small generating set for the balanced superelliptic handlebody group

The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We prove that the balanced superelliptic mapping class group is generated by four elements. To prove this, we also proved that the liftable Hilden group is generated by three elements. This generating set for the liftable Hilden group is minimal except for some hyperelliptic cases and the generating set for the balanced superelliptic mapping class group above is also minimal for several cases.