Action of mapping class groups and its subgroups

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.

我的研究在于低维拓扑和几何群体理论的领域，特别是Riemann表面，映射课程组和Teichmueller理论。我一直有兴趣通过在Teichmueller空间上的动作以及从曲线复合物，ARC复合物或翻转图等表面上构建的某些简单络合物来理解映射类组及其子组。最近，我在动力学中采用了这些技术，并开始研究仿射组对通过翻译表面鞍座建立的简单复合物的作用。我目前的研究有三个轴，与该项目的三个部分相对应：I.映射类组和刚性的组合作用； ii。 （多）弧的复合物的大规模几何以及几何组理论与动力学之间的相互作用； iii。 Thurston在Teichmueller空间上的距离及其在高级Teichmueller理论中的概括。第一部分涉及映射课程组和简单的刚性问题。在1990年代，伊万诺夫（Ivanov）证明了映射类组可以表示为曲线复合体的自动形态组。随后，与表面相关的许多其他简单络合物都暴露了相同的特征。了解所有这些对象的共同点仍然是一个开放的问题（伊万诺夫的元调查）。最近，布伦德·马格利特（Brendle-Margalit）用这种属性来表征一个整个复合物，并在这方面提出了一个精确的概念。我正在针对多弧线攻击这个问题，并研究曲线复杂的模型理论以及与Thomas Koberda和Javier de la Nuez-Gonzalez一起绘制的类群相关的其他图表。第二部分介绍了弧形复合物，翻转图，相关算法问题以及这些工具在动力学中的新应用。在Masur-Minsky的基础作品之后，人们对曲线复合物和其他模拟组合复合物的大规模特性引起了很多兴趣。几何群体理论的最新趋势是研究弧和三角形的复合物。这些物体在其他数学领域（理论计算机科学，集群代数，量子拓扑，Margulis Space Times ...）中具有更多的动机。我有兴趣通过三角形图理解映射类组的大规模几何形状。我正在使用Anja Randecker和Robert Tang开发一个程序，从动力学的几何组理论到员工技术，通过研究由翻译表面上的马鞍连接构建的简单复合物的研究。第三部分涉及Thurston在Teichmueller空间上的距离。 Thurston的距离是Thurston定义的Teichmueller距离的类似物。与Daniele Alessandrini一起，我们将Thurston的结果推广到具有边界的表面。