具有非紧边界曲面的映射类群

A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

We prove that each Torelli group of an orientable 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 each Torelli group in the level $d$ mapping class group is the same as that of in the mapping class group.

An infinite presentation for the mapping class group of a non-orientable surface with boundary

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

Irreducibility of quantum representations of mapping class groups with boundary

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.

Mapping Class Group Dynamics on Surface Group Representations

Deformation spaces Hom($π$,G)/G of representations of the fundamental group $π$ of a surface $Σ$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_Σ$, preserving a Poisson structure. When $G$ is compact, the actions are ergodic. In contrast if $G$ is noncompact semisimple, the associated deformation space contains open subsets containing the Fricke-Teichmüller space upon which $Mod_Σ$ acts properly. Properness of the $Mod_Σ$-action relates to (possibly singular) locally homogeneous geometric structures on $Σ$. We summarize known results and state open questions about the…

Geometry of the mapping class groups I: Boundary amenability

We construct a geometric model for the mapping class group M of a non-exceptional oriented surface of finite type and use it to show that the action of M on the compact Hausdorff space of complete geodesic laminations is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of M.

Generating mapping class groups of nonorientable surfaces with boundary

We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.

曲面的端和

The end sum of surfaces

本文研究曲面上的端和及在无穷远处添加1-柄，证明其唯一性由所选端和可定向性决定，并依赖非紧曲面分类。

A new boundary of the mapping class group

Based on the action of the mapping class group on the space of measured foliations, we construct a new boundary of the mapping class group and study the structure of this boundary. As an application, for any point in Teichmuller space, we consider the orbit of this point under the action of the mapping class group and describe the closure of this orbit in the Thurston compactification and the Gardiner-Masur compactification of Teichmuller space. We also construct some new points in the Gardiner-Masur boundary of Teichmuller space.

A Thurston boundary for infinite-dimensional Teichmüller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T…

The first homology group with twisted coefficients for the mapping class group of a non-orientable surface of genus three with two boundary components

We determine the first homology group with coefficients in $H_1(N;\mathbb{Z})$ for the mapping class group of a non-orientable surface $N$ of genus three with two boundary components.

Boundary actions of lattices and $C^0$ local semi-rigidity

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate t…

On TQFT representations of mapping class groups with boundary

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.

带边界不可定向曲面映射类群的扭曲系数第一同调群

The first homology group with twisted coefficients for the mapping class group of a non-orientable surface with boundary

本文计算了带穿孔和/或边界的不可定向曲面各类映射类群在H_1(N;Z)系数下的第一同调群。

带边流形的映射类群是有限型的

Mapping class groups of manifolds with boundary are of finite type

本文证明在适当的维数与连通性条件下，带边紧致拓扑流形的映射类群具有有限型。

带边界双曲曲面之间的双测地映射

Bi-geodesic mappings between hyperbolic surfaces with boundary

该研究证明带边界紧双曲曲面间满足自身与逆映射都保测地的双射必为等距映射，完善了双曲曲面测地映射的相关理论。

Homology of the mapping class group for surfaces of genus 2 with boundary

We report on the computation of the integral homology of the mapping class group of genus g surfaces with one boundary curve and m punctures, when 2g + m is smaller than 6. In particular, it includes the genus 2 case with no or one puncture.