Some problems on mapping class groups and moduli space

Combinatorial analysis of the period mapping: topology of 2D fibers

We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of the moduli space into polyhedra labeled by planar graphs allows to investigate the global topology of low dimensional fibers of the periods mapping.

The Torelli group and congruence subgroups of the mapping class group

These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers "The second rational homology group of the moduli space of curves with level structures" and "The Picard group of the moduli space of curves with level structures".

高维流形的h-同调簇群

Mapping class groups of $h$-cobordant manifolds

本文通过构造h-同调流形的映射类群差异，证明高维流形的映射类群不是h-同调不变量。

黎曼曲面与柄体模空间的最高权上同调

Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies

本研究证明双曲曲面模空间$M_g$中含足够多短测地线的曲面轨迹是柄体映射类群的分类空间，由此推出$M_g$的最高权上同调可单射到柄体映射类群的上同调中。

Homological action of the modular group on some cubic moduli spaces

We describe the action of the automorphism group of the complex cubic x^2+y^2+z^2-xyz-2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus on the subvarieties of its SL(2,C) character variety given by fixing the trace of the peripheral element (so-called "relative character varieties"). This mapping class group is isomorphic to PGL(2,Z). We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2,C) character varieties, which are fibers of deformations x^2+y^2+z^2-xyz-2-Px-Qy-Rz of the…

Probing mapping class groups using arcs

The action of the mapping class group of a surface on the collection of homotopy classes of disjointly embedded curves or arcs in the surface is discussed here as a tool for understanding Riemann's moduli space and its topological and geometric invariants. Furthermore, appropriate completions, elaborations, or quotients of the set of all such homotopy classes of curves or arcs give for instance Thurston's boundary for Teichmüller space or a combinatorial description of moduli space in terms of fatgraphs. Related open problems and questions are discussed.

The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II

We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simply-connected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, …

Torsion functions on moduli spaces in view of the cluster algebra

We introduce non-acyclic $PGL_n(\mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's $PGL_2(\mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(\mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichmüler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.

The second rational homology group of the moduli space of curves with level structures

Let $Γ$ be a finite-index subgroup of the mapping class group of a closed genus $g$ surface that contains the Torelli group. For instance, $Γ$ can be the level $L$ subgroup or the spin mapping class group. We show that $H_2(Γ;\Q) \cong \Q$ for $g \geq 5$. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\Q$. We also prove analogous results for surface with punctures and boundary components.

扭曲全纯1-形式的模空间与膨胀曲面分层上的不变测度

Invariant measures on moduli spaces of twisted holomorphic 1-forms and strata of dilation surfaces

针对黎曼曲面上扭曲全纯1-形式的模空间，通过映射类群系数上同调的新颖计算，构造了分层上的SL(2,R)-不变Lebesgue类测度，即Masur-Veech测度的类似物。

Counting closed geodesics in Moduli space

We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.

The geometry of right angled Artin subgroups of mapping class groups

We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space.

Global rigidity of the period mapping

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and $h\leq g$. If $F:{\mathcal M}_{g,n}\to{\mathcal A}_h$ is a nonconstant holomorphic map then $h=g$ and $F$ is the classical period mapping, assigning to a Riemann surface $X$ its Jacobian.

Surface group representations in ${\rm SL}_2({\mathbb C})$ with finite mapping class orbits

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.

映射类群在奇数特征簇上的作用是忠实的

The mapping class group action on the odd character variety is faithful

本文证明亏格至少2时映射类群有限扩张在奇数特征簇及其Fukaya范畴上的作用忠实，推广Smith结果并回答Dostoglou-Salamon问题。

Strings and the stable cohomology of mapping class groups

Twenty years ago, Mumford initiated the systematic study of the cohomology ring of moduli spaces of Riemann surfaces. Around the same time, Harer proved that the homology of the mapping class groups of oriented surfaces is independent of the genus in low degrees, increasing with the genus. The (co)homology of mapping class groups thus stabelizes. At least rationally, the mapping class groups have the same (co)homology as the corresponding moduli spaces. This prompted Mumford to conjecture that the stable rational cohomology of moduli spaces is generated by certain tautological classes that he …

Comparing fat graph models of moduli space

Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homolog…

Picard groups of moduli spaces of curves with symmetry

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the second part of the paper, we show that finite abelian level covers of the hyperelliptic locus …

The action of the mapping class group on representation varieties of PSL(2,R). Case I: The one-holed torus

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the o…

Lectures and notes: Mirzakhani's volume recursion and approach for the Witten-Kontsevich theorem on moduli tautological intersection numbers

The materials accompany a lecture short course presented at the 2011 Park City Mathematics Institute, Graduate Summer School on Moduli Spaces of Riemann Surfaces. The lectures were part of/coordinated with an overall program, including lectures by Ursula Hamenstadt on Teichmueller Theory, Andy Putman on Mapping Class and Torelli Groups, and Carel Faber on Tautological algebras of Moduli Spaces.

Tessellating the moduli space of strictly convex projective structures on the once-punctured torus

We show that associating the Euclidean cell decomposition due to Cooper and Long to each point of the moduli space of framed strictly convex real projective structures of finite volume on the once-punctured torus gives this moduli space a natural cell decomposition. The proof makes use of coordinates due to Fock and Goncharov, the action of the mapping class group as well as algorithmic real algebraic geometry. We also show that the decorated moduli space of framed strictly convex real projective structures of finite volume on the thrice-punctured sphere has a natural cell decomposition.

Orbifold Kähler Groups related to Mapping Class groups

We construct certain orbifold compactifications of the moduli stack of pointed stable curves over $\mathbb C$ and study their fundamental groups by means of their quantum representations. This enables to construct interesting Kähler groups and to settle most of the candidates for a counter-example to the Shafarevich conjecture on holomorphic convexity proposed in 1998 by Bogomolov and Katzarkov, using TQFT representations of the mapping class groups.

An analytic construction of the Deligne-Mumford compactification of the moduli space of curves

In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.

Moduli spaces of complex affine and dilation surfaces

We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech, we show that the the moduli space of affine surfaces with fixed genus and with cone points of fixed complex order is a holomorphic affine bundle over the moduli space of Riemann surfaces. Similarly, the moduli space of dilation surfaces is a covering space of the moduli space of Riemann surfaces. We classify the connected components of the moduli space of dilation surfaces and show that any component is an orbifold K(G,1) where G is the framed mapping class group of Calderon-Salter.

轨形上平坦黎曼度量的模空间

Moduli spaces of flat Riemannian metrics on orbifolds

本文研究闭黎曼轨形的平坦度量模空间，证明其 Teichmüller 空间可分类映射类群的真作用，且模空间本身是非常好的轨形，并在技术假设下给出其到可公度性的代数描述。

Higher spin mapping class groups and strata of Abelian differentials over Teichm{ü}ller space

For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the non-hyperelliptic components are classified by an invariant known as an $r$-spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all $r$-spin stabilizer subgroup…

映射类群的代数几何子群

Algebro-geometric subgroups of mapping class groups

本文通过约束映射类群的代数几何子群（即光滑曲线模空间上复代数映射下曲线基本群的像），证明当亏格至少为3时，闭曲面映射类群的无限秩有限秩酉表示在代数几何子群上的限制必为无限维，并由此推出大多数此类紧化的万有覆盖是Stein流形。

Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmueller space. The main tool for these results is the …

Representation stability for the cohomology of the moduli space M_g^n

Let M_g^n be the moduli space of Riemann surfaces of genus g with n labeled marked points. We prove that, for g \geq 2, the cohomology groups {H^i(M_g^n;Q)}_{n=1}^{\infty} form a sequence of Sn representations which is representation stable in the sense of Church-Farb [CF]. In particular this result applied to the trivial Sn representation implies rational "puncture homological stability" for the mapping class group Mod_g^n. We obtain representation stability for sequences {H^i(PMod^n(M);Q)}_{n=1}^{\infty}, where PMod^n(M) is the mapping class group of many connected manifolds M of dimension d…

Groupoid Extensions of Mapping Class Representations for Bordered Surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolem…

Teichmuller geometry of moduli space, II: M(S) seen from far away

We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.

Coordinates on the augmented moduli space of convex RP^2 structures

Let S be an orientable, finite type surface with negative Euler characteristic. The augmented moduli space of convex real projective structures on S was first defined and topologized by the first author. In this article, we give an explicit description of this topology using explicit coordinates. More precisely, given every point in this augmented moduli space, we find explicit continuous coordinates on the quotient of a suitable open neighborhood about this point by a suitable subgroup of the mapping class group of S. Using this, we give a simpler proof of the fact that the augmented moduli s…

双曲曲面模空间上一般几何随机变量的积分

Integrals of general geometric random variables on the moduli space of hyperbolic surfaces

本文推广了Mirzakhani的简单测地线积分公式，给出了双曲曲面模空间上涉及任意固定映射类群轨道中闭测地线长度的随机变量的积分公式，并利用该公式改进了Weil-Petersson期望的渐近等价。

On the topology of moduli spaces of real algebraic curves

We show that mapping class groups associated to all types of real algebraic curves are virtual duality groups. We also deduce some results about the orbifold homotopy groups of the moduli spaces of real algebraic curves. We achieve these results by defining a new complex associated to a not necessarily orientable surface with boundary called the $\mathcal{A}\mathcal{B}\mathcal{C}$-complex. This complex encodes the intersection patterns of isotopy classes of essential simple arcs, boundary components, and essential simple closed curves. We show that this complex is homotopy equivalent to an inf…

Flip-graph moduli spaces of filling surfaces

This paper is about the geometry of flip-graphs associated to triangulations of surfaces. More precisely, we consider a topological surface with a privileged boundary curve and study the spaces of its triangulations with n vertices on the boundary curve. The surfaces we consider topologically fill this boundary curve so we call them filling surfaces. The associated flip-graphs are infinite whenever the mapping class group of the surface (the group of self-homeomorphisms up to isotopy) is infinite, and we can obtain moduli spaces of flip-graphs by considering the flip-graphs up to the action of…

The geometry of flip graphs and mapping class groups

The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichmüller space and is quasi-isometric to the underlying mapping class group. We study this space in two main directions. We first show that strata corresponding to triangulations containing a same multiarc are strongly convex within the whole space and use this result to deduce properties about the mapping class group. We then focus on the quotient of this space by the mapp…

Universality theorems for configuration spaces of planar linkages

We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.

Homological stability for moduli spaces of high dimensional manifolds. I

We prove a homological stability theorem for moduli spaces of simply-connected manifolds of dimension $2n > 4$, with respect to forming connected sum with $S^n \times S^n$. This is analogous to Harer's stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of the moduli spaces of manifolds diffeomorphic to connected sums of $S^n \times S^n$ in a range of degrees.

关于带孔曲面的 $SU(2)$ 特征簇的辛方面

On symplectic aspects of $SU(2)$ character varieties for punctured surfaces

本文研究带奇数孔曲面的 $SU(2)$ 平坦联络模空间的辛性质，证明映射类群到辛映射类群的同态在维数大于2时是单射，并给出5点爆破射影平面中拉格朗日球面的完全分类及其可移动性判定。

代数映射类群刚性

Algebraic Mapping Class Group Rigidity

研究关于具有\(n\)个穿孔的亏格为\(g\)的连通定向曲面\(\Sigma\)，利用\(\mathrm{SL}_2\)特征标簇的刚性相对自同构群及相关特殊同构，对\(\Sigma\)的映射类群给出新描述。

The tropicalization of the moduli space of curves II: Topology and applications

We study the topology of the tropical moduli space parametrizing stable tropical curves of genus g with n marked points in which the bounded edges have total length 1, and prove that it is highly connected. Using the identification of this space with the dual complex of the boundary in the moduli space of stable algebraic curves, we give a simple expression for the top weight cohomology of M_{1,n} as a representation of the symmetric group and describe an explicit dual basis in homology consisting of abelian cycles for the pure mapping class group.

Random veering triangulations are not geometric

Every pseudo-Anosov mapping class $φ$ defines an associated veering triangulation $τ_φ$ of a punctured mapping torus. We show that generically, $τ_φ$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

Hitchin-Riemann模空间的体积是无限的

The volumes of the Hitchin-Riemann moduli spaces are infinite

本文证明映射类群在秩至少为二的高阶Teichmüller空间上的作用具有无限Atiyah-Bott-Goldman余体积，涵盖G-Hitchin分量与Sp(2n,R)-极大表示空间分量，方法上借助Goldman流构造无限多个等体积且商空间中互不相交的子集。

Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

We extend Teichmueller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. We observe how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the derormation space and of each stratum of the Teichmueller unit sphere bundle over the Riemann moduli space. We prove ergodicity for the analogous…

Classification of Rauzy classes in the moduli space of quadratic differentials

We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.

Structure of the mapping class groups of surfaces: a survey and a prospect

In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3-manifolds, the unstable cohomology of the moduli space of curves and Faber's conjecture, cokernel of the Johnson homomorphisms and the Galois as well as other new obstructions, cohomology of certain infinite dimensional Lie algebra and characteristic classes of outer automorphism groups of free group…

Random Lattices, Punctured Tori and the Teichmüller distribution

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teichmüller space to the central "square" punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichmüller space to the rectangular punctured tor…

4-流形模空间的同调可能无限生成

The homology of moduli spaces of 4-manifolds may be infinitely generated

针对单连通闭流形映射类群有限生成在4维失效的问题，证明存在单连通闭光滑4-流形的映射类群非有限生成，且对任意$k>0$，存在4-流形使$H_k(B\text{Diff}(X);\text{Z})$非有限生成，贡献了4维流形模空间同调的新结果。

扭曲的局部野映射类群：配置空间、分裂树和复杂编织

Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

本文研究了类型A的任意形式结构下的野映射类群，通过构造配置空间和分裂森林，揭示了复杂反射群作为野映射类群的出现，并计算了野黎曼曲面模空间的维度。

On the cohomology of pure mapping class groups as FI-modules

In this paper we apply the theory of finitely generated FI-modules developed by Church, Ellenberg and Farb to certain sequences of rational cohomology groups. Our main examples are the cohomology of the moduli space of n-pointed curves, the cohomology of the pure mapping class group of surfaces and some manifolds of higher dimension, and the cohomology of classifying spaces of some diffeomorphism groups. We introduce the notion of FI[G]-module and use it to strengthen and give new context to results on representation stability discussed by the author in a previous paper. Moreover, we prove tha…

Asymptotic properties of the quantum representations of the mapping class group

We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that the operators in these representations are Fourier integral operators and determine explicitly their canonical relations and symbols. We deduce from these facts the Egorov property and the asymptotic unitarity, two results already proved by J.E. Andersen. Furthermore we show und…