Action of mapping class groups and its subgroups

我的研究主要集中在低维拓扑和几何群论领域，特别是Riemann曲面、映射类群和Teichmueller理论。我一直有兴趣通过映射类群及其子群在Teichmueller空间和由曲面上的拓扑对象(如曲线复形、弧复形或翻转图)建立的某些单纯复形上的作用来理解映射类群及其子群。最近，我在动力学中应用了这些技巧，并开始研究仿射群在平移曲面上由鞍形连接建立的单纯复形上的作用。我目前的研究有三个轴，它们对应于这个项目的三个部分：一、映射类群与刚性的组合作用；二、(多)弧复的大规模几何以及几何群论与动力学之间的相互作用；三、瑟斯顿在Teichmueller空间上的距离及其在高等Teichmueller理论中的推广。第一部分讨论了映射类群和单纯刚性问题。20世纪90年代，Ivanov证明了映射类群可以表示为曲线复形的自同构群。随后，许多与曲面相关的其他单纯复形也表现出同样的特征。理解所有这些物体的共同点仍然是一个悬而未决的问题(伊万诺夫的元猜想)。最近，Brendle-Margalit刻画了一整族具有这种性质的络合物，并在这方面提出了一个精确的猜想。我和Thomas Koberda和Jille de la Nuez-Gonzalez一起研究了多弧的这个问题，并研究了曲线复合体的模型理论和其他与映射类群相关的图。第二部分讨论了弧形复形、翻转图的几何群论、相关的算法问题以及这些工具在动力学中的新应用。在Masur-Minsky的基础工作之后，人们对曲线复形和其他类似的组合复形的大规模性质产生了很大的兴趣。几何群论的一个最新趋势是研究弧形和三角剖分的复合体。进一步的动机来自这些对象在其他数学领域的许多应用(理论计算机科学、集群代数、量子拓扑学、马古利斯空间时间……)。我感兴趣的是通过三角剖分图来理解映射类组的大尺度几何。我正在与Anja Randecker和Robert Tang一起开发一个程序，通过研究平移曲面上的鞍形连接建立的单纯复形，使用动力学中几何群论的技术。第三部分讨论了瑟斯顿在Teichmueller空间上的距离。瑟斯顿距离是瑟斯顿定义的泰希穆勒距离的类比。与Daniele Alessandrini一起，我们将瑟斯顿的结果推广到带边界的曲面。

My research lies in the fields of low-dimensional topology and geometric group theory, in particular Riemann surfaces, mapping class groups and Teichmueller theory. I have been interested in understanding the mapping class group and its subgroups via their actions on the Teichmueller space and on certain simplicial complexes built from topological objects on the surfaces such as the curve complex, the arc complex or the flip graph. Recently I have employed these techniques in dynamics and started investigating the actions of the affine group on simplicial complexes built from saddle connections on a translation surface. My current research has three axis, which correspond to the three parts of this project: I. combinatorial actions of the mapping class group and rigidity; II. large scale geometry of complexes of (multi-)arcs and interactions between geometric group theory and dynamics; III.Thurston’s distance on the Teichmueller space and its generalizations in higher Teichmueller theory. Part I deals with mapping class groups and the simplicial rigidity problem. In the 1990s Ivanov proved that the mapping class group can be represented as the automorphism group of the curve complex. Subsequently, many other simplicial complexes associated to a surface have exhibited this same feature. Understanding what all these objects have in common is still an open problem (Ivanov’s metaconjecture). Recently Brendle-Margalit characterize one entire family of complexes with this property and formulated a precise conjecture in this regard. I am attacking this problem for multi-arcs and studying the model theory of the curve complex and other graphs associated to mapping class groups together with Thomas Koberda and Javier de la Nuez-Gonzalez. Part II deals with the geometric group theory of arc complexes, flip graphs, related algorithmic problems and a new application of these tools in dynamics. After foundational works by Masur-Minsky there has been a lot of interest in the large scale properties of the curve complex and other analogue combinatorial complexes. A recent trend in geometric group theory is to look at complexes of arcs and triangulations. Further motivations come from the many applications these objects have in other fields of mathematics (theoretical computer science, cluster algebras, quantum topology, Margulis space times...). I am interested in understanding the large scale geometry of the mapping class group via graphs of triangulations. I am developing a program with Anja Randecker and Robert Tang to employ techniques from geometric group theory in dynamics via the study of simplicial complexes built from saddle connections on a translation surface. Part III deals with Thurston’s distance on Teichmueller space. Thurston’s distance is an analogue of Teichmueller distance defined by Thurston. Together with Daniele Alessandrini we are generalizing Thurston’s results to surfaces with boundary.