L. Paris
We prove in this paper that the action of the mapping class group on the complex of curves has noncommensurable stabilizers. Following a method due to Burger and de la Harpe, this action leads to constructions of irreducible unitary representations of the mapping class group.
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We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Σ;\mathbb{Z})$.
We prove that for every P there is a bound B depending only on P so that the mapping torus of every P--small irreducible train-track map can be obtained by surgery from one of B mapping tori. We show that given an integer P>0 there is a bound $M$ depending only on P, so that there exists a presentation of the fundamental group of the mapping torus of a P--small irreducible train-track map with less than M generators and M relations.
Given an irreducible, end-periodic homeomorphism f of a surface S with finitely many ends, all accumulated by genus, the mapping torus is the interior of a compact, irreducible, atoroidal 3-manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of the compactified mapping torus in terms of the translation length of f on the pants graph of S. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotical…
Teichmüller isometries induced by certain irreducible periodic mapping classes
本文提出了一种算法，用于在Fenchel-Nielsen坐标下确定由某些不可约周期映射类诱导的Teichmüller空间等距，并以4g+2阶周期映射类为例给出了描述和几何性质。
We find decomposition series of length at most two for modular representations in positive characteristic of mapping class groups of surfaces induced by an integral version of the Witten-Reshetikhin-Turaev SO(3)-TQFT at the p-th root of unity, where p is an odd prime. The dimensions of the irreducible factors are given by Verlinde-type formulas.
Let $Γ$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $Γ$ on a $\CAT(0)$ cubical complex has a global fixed point.
We prove that the Witten--Reshetikhin--Turaev $\mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.
On the action of relatively irreducible automorphisms on their train tracks
本文研究相对𝒢的指数型𝒢-不可约自同构的轨道轨点集（即最小位移集Min(φ)），证明其在φ生成的循环子群作用下余紧，得到相关辅助结果，并推广了自由群自同构中心化子的经典定理。
We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the cir…
Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) 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 …
Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, we show some applications of our an…
A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.
Graphs of curves on infinite-type surfaces with mapping class group actions
该研究探讨无限型曲面映射类群在简单闭曲线连通图上有无界轨道作用的条件，引入新拓扑不变量判定该作用存在性，证明诸多非局部紧的大映射类群具有Rosendal意义下的非平凡粗几何性质。
The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.
We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.
We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitel…
We analyze the mapping class group of extendible automorphisms of the exterior boundary W of a compression body of dimension 3 or 4, which extend over the compression body (Q,V), where V is the interior boundary. Those that extend as automorphisms of (Q,V) rel V are called discrepant automorphisms, forming the mapping class group of discrepant automorphisms of W in Q. We describe a short exact sequence of these mapping class groups. For an orientable, compact, reducible 3 manifold W, there is a canonical "maximal" 4-dimensional compression body Q whose exterior boundary is W and whose interior…
We obtain representations of the f-based Mapping Class Group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.
We study the TQFT mapping class group representations for surfaces with boundary associated with the $SU(2)$ gauge group, or equivalently the quantum group $U_q(\Sl(2))$. We show that at a prime root of unity, these representations are all irreducible. We also examine braid group representations for transcendental values of the quantum parameter, where we show that the image of every mapping class group is Zariski dense.
For every irreducible automorphism $ϕ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $ϕ$, whose stretch factor is the product of the expanding eigenvalues of $ϕ$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $ϕ$ has exactly two eigenvalues outside the unit cir…
Let $Σ$ be a complete finite-area orientable hyperbolic surface with one cusp, and let $\mathcal{R}$ be the space of complete geodesic rays in $Σ$ emanating from the puncture. Then there is a natural action of the mapping class group of $Σ$ on $\mathcal{R}$. We show that this action is "almost everywhere" wandering.
S. Gervais gave a finite presentation for the mapping class group of a surface (math.GT/9811162). We show this presentation without using Wajnryb's simple presentation. We use a complex of curves defined by Harvey, in place of Hatcher and Thurston's complex.
This paper has three parts. The first part is a general introduction to rigidity and to rigid actions of mapping class group actions on various spaces. In the second part, we describe in detail four rigidity results that concern actions of mapping class groups on spaces of foliations and of laminations, namely, Thurston's sphere of projective foliations equipped with its projective piecewise-linear structure, the space of unmeasured foliations equipped with the quotient topology, the reduced Bers boundary, and the space of geodesic laminations equipped with the Thurston topology. In the third …
We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. We also study action of the mapping class group of the surface on its Fukaya category and establish faithfulness.
The mapping class group $\mathrm{Mod}_{g, 1}$ of a surface with one marked point can be identified with an index two subgroup of $\mathrm{Aut}(π_1 Σ_g)$. For a surface of genus $g \geq 2$, we show that any action of $\mathrm{Mod}_{g, 1}$ on the circle is either semi-conjugate to its natural action on the Gromov boundary of $π_1 Σ_g$, or factors through a finite cyclic group. For $g \geq 3$, all finite actions are trivial. This answers a question of Farb.
We study the action of the mapping class group M(F) on the complex of curves of a non-orientable surface F. We obtain, by using a result of K. S. Brown, a presentation for M(F) defined in terms of the mapping class groups of the complementary surfaces of collections of curves, provided that F is not sporadic, i.e. the complex of curves of F is simply connected. We also compute a finite presentation for the mapping class group of each sporadic surface.
The action of mapping class groups on de Rham quasimorphisms
本文研究映射类群对双曲曲面二次有界上同调中 de Rham 类的作用，证明唯一不变的非平凡有限维子空间由 Euler 类生成，并推出其在 de Rham 拟态射空间上的作用无不动点。
We give a short and elementary proof of the non-realizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.
The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be 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. given by the pushforward of Haar measure. This result is an extension of earlier results of Goldman for orientable surfaces.
We prove the following well known conjecture: let $Σ$ be an oriented surface of finite type whose fundamental group is a nonabelian free group. Let $ϕ\in \textup{Mod}(Σ)$ be a an infinite order mapping class. Then there exists a finite solvable cover $\widehatΣ \to Σ$, and a lift $\widehatϕ$ of $ϕ$ such that the action of $\widehatϕ$ on $H_1(\widehatΣ, \mathbb{Z})$ has infinite order. Our main tools are the theory of homological shadows, which was previously developed by the author, and Fourier analysis
We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüll…
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.
This note investigates the dynamics of the mapping class group action on the character variety of super-maximal representations of the fundamental group of a punctured sphere into $\text{PSL}(2,\mathbb{R})$, discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.
We give completely combinatorial proofs of the main results of [3] using polygons. Namely, we prove that the mapping class group of a surface with boundary acts faithfully on a finitely-generated linear category. Along the way we prove some foundational results regarding the relevant objects from bordered Heegaard Floer homology,
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.
Mapping class group action on the cohomology of the $\mathrm{SL}_n$ character variety
研究映射类群对曲面$\Sigma_g$扭曲$\mathrm{SL}_n$特征流形上同调的作用，利用Maulik-Shen的相对内核分解，将问题转化为对$\Sigma_g$的canonical有限覆盖的同调作用。
We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin^c-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.
Let X_n be a cycle of n projective lines, and T_n a symplectic torus with n punctures. In this paper we review results appeared in arXiv:1103.2462 and in arXiv:1109.6615, which establish a version of homological mirror symmetry relating X_n and T_n, and define on D^b(Coh(X_n)) an action of the pure mapping class group of T_n.
A modular fusion category C allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then C has a unique Morita-class of simple non-degenerate algebras, namely that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.
Let $X_n$ be a cycle of $n$ projective lines, and $\bT_n$ a symplectic torus with $n$ punctures. Using the theory of spherical twists introduced by Seidel and Thomas (2001), I will define an action of the pure mapping class group of $\bT_n$ on $D^b(Coh(X_n))$. The motivation comes from homological mirror symmetry for degenerate elliptic curves, which was studied by the author with Treumann and Zaslow in arXiv:1103.2462 .
We present a rigidity theorem for the action of the mapping class group $π_0(\mathrm{Diff}(M))$ on the space $\mathcal{R}^+(M)$ of metrics of positive scalar curvature for high dimensional manifolds $M$. This result is applicable to a great number of cases, for example to simply connected $6$-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the $2$-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected $7$-dimensional $\mathr…
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