Geometric Group Theory

这个项目将集中在两个重要的几何类群：映射类群的表面;和外部自同构群的自由团体。 利用大尺度几何的工具研究映射类群，目的是了解映射类群的拟等距性质并证明其拟等距刚性，沿着这条道路，我们期望发展新的工具来理解相对双曲群的拟等距性质。 外自同构群不像映射类群那样被很好地理解，并且将从更基础的角度进行研究。 重点将是发展与Teichmuller空间理论的类比，这是一个非常发达的工具，用于研究映射类群，特别是发展与最后层合猜想的类比。几何群论是研究与无限范围的高度对称几何相关的对称群，例如三维空间中的晶格，它的对称群是所有空间运动的集合，这些运动将格子中的每一点移动到格子中的另一点。 群论是在世纪为研究对称性而发展起来的，并迅速发展成为一种深刻而丰富的数学工具。 组和几何是紧密联系的：每个组都有相应的几何，人们可以使用几何工具来了解组。 这个项目将集中在两个类的群体已承担的重要性领域的拓扑结构。 给定一个表面，比如有一个或多个孔的甜甜圈表面，该表面的“映射类组”描述了该表面可以映射回自身的所有方式。 类似地，“自由群的外自同构群”是描述有限一维网络或图可以映射回自身的所有方式的群。 在这两种情况下，我们将研究与这些群体相关的几何形状，目的是更多地了解这些群体本身。 在映射类群的情况下，我们的目的是证明一个中心几何定理，称为拟等距刚性。 在自由群的外自同构群的情况下，几何学还不太清楚，我们将尝试建立新的工具，并对现有的工具进行更深入的研究，以便更好地理解几何学。

This project will focus on the geometry of two important classes ofgroups: mapping class groups of surfaces; and outer automorphism groupsof free groups. Mapping class groups will be studied using the tools oflarge scale geometry, with the aim of learning about quasi-isometricproperties of mapping class groups and proving quasi-isometric rigidity.Along the way we expect to develop new tools for understandingquasi-isometric properties of relatively hyperbolic groups. Outerautomorphism groups are not as well understood as mapping class groups,and will be studied from a more foundational point of view. The focuswill be to develop analogies with the theory of Teichmuller spaces whichis a very well developed tool for studying mapping class groups, with theparticular aim of developing analogies with the ending laminationconjecture.Geometric group theory is the study of symmetry groups associated tohighly symmetric geometries of infinite extent, such as a crystal latticein 3-dimensional space, whose symmetry group is the set of all motions ofspace which move each point of the lattice to another point of thelattice. The theory of groups was developed in the 19th century forstudying symmetry, and it quickly developed into a deep and richmathematical tool. Groups and geometries go hand-in-hand: every group hasa corresponding geometry, and one can use tools of geometry to learn aboutthe group. This project will focus on two classes of groups which haveassumed importance in the field of topology. Given a surface, such as thesurface of a doughnut with one or more holes, the "mapping class group" ofthat surface describes all the ways in which that surface can be mappedback onto itself. Similarly, the "outer automorphism group of a freegroup" is a group that describes all of the ways in which a finite1-dimensional network or graph can be mapped back onto itself. In bothcases, we will study geometries associated to these groups, with the aimof learning more about the groups themselves. In the case of mappingclass groups our aim is to prove one of the central geometric conjectures,known as quasi-isometric rigidity. In the case of outer automorphismgroups of free groups, where the geometry is less well understood, we willattempt to build new tools and make a deeper study of existing tools, inorder to reach a better understanding of the geometry.