Geometric Group Theory

摘要奖：DMS-0103208首席研究员：Lee Mosher几何群论的启发主题是有限生成的群可以用拓扑学和几何学的方法来研究。从1980年开始，Gromov提出了利用拟等距关系对群进行几何分类。他指出，群G的拟等距类往往可以用简单的代数或几何术语来明确地描述，这个过程现在被称为G的“拟等距刚性”或“QI-刚性”。这个项目的第一部分将研究表面群和交换群的图的QI-刚性问题。最初的焦点将是具有最简单的代数结构的群的图，即具有自由群的半直积，由从自由群到适当的自同构群的同态确定，例如曲面的映射类群。曲面群的一般图是由到可度量映射类群的同态确定的，这是一个更神秘的问题，这一观点将被用来研究新的和有趣的例子的构造。此外，最近对交换群的图的工作揭示了许多丰富的结构，这表明在许多新的情况下获得QI-刚性是真实的可能性。在这个项目的第二部分，重点将是研究自由群的几何性质，其动机是表面群和自由群之间的类比。特别是，瑟斯顿的终结分层猜想，表面群研究中的一个重要目标，在自由群的研究中有一个相似之处，就是将某些群作用归类为等变准等距。追求这一问题将要求曲面群的许多标准工具，如TeichMuller空间中的测地线，重新适用于自由群的设置。几何群论是对无限对称模式的研究。从墙纸的对称性和埃舍尔的印刷品的对称性中，人们熟悉被称为‘表面群’的流行例子。科学的例子出现在晶体阵列的对称群中，以及粒子物理学中场论的对称群中。开始于19世纪末的拓扑学的发展，以及随之而来的组合群论的发展，显示了抽象群和几何之间的直接联系。在20世纪的数学发展中，不同的线索一次又一次地证明了深入理解这一联系的必要性。其中许多线索是由格罗莫夫在1980年前后联系在一起的，他提出的利用“准等距”关系统一几何群论的建议，在其间的20年里取得了非常丰硕的成果。本研究项目的重点是研究几种不同类型对称群的拟等距分类问题。具体地说，通过使用一种名为“群的图”的构造性技术，可以从熟悉的例子(如表面群)中构造新的对称群；这些及其密切相关的构造将是本研究项目的主题。

AbstractAward: DMS-0103208Principal Investigator: Lee MosherThe motivating theme of geometric group theory is that finitelygenerated groups can be studied using topological and geometricmethods. Starting around 1980, Gromov proposed classifyinggroups geometrically using the relation of quasi-isometry. Hedemonstrated that the quasi-isometry class of a group G can oftenbe described explicitly in simple algebraic or geometric terms, aprocess now referred to as ``quasi-isometric rigidity'' or``QI-rigidity'' for the group G. The first part of this projectwill be to investigate QI-rigidity problems for graphs of surfacegroups and of abelian groups. An initial focus will be thosegraphs of groups with the simplest algebraic structure, namelysemidirect products with free groups, determined by ahomomorphism from a free group into an appropriate automorphismgroup such as the mapping class group of a surface. Generalgraphs of surface groups are determined by homomorphisms into thecommensurability mapping class group, a much more mysteriousobject, and this point of view will be used to investigateconstructions of new and interesting examples. Also, recent workon graphs of abelian groups has revealed a lot of rich structure,suggesting a real possibility of obtaining QI-rigidity in manynew cases. In the second part of this project, the focus will beto study geometric properties of free groups, motivated byanalogies between surface groups and free groups. In particular,Thurston's ending lamination conjecture, an important goal in thestudy of surface groups, has an analogue in the study of freegroups, in terms of classifying certain group actions up toequivariant quasi-isometry. Pursuing this issue will requiregeneralizing many of the standard tools of surface groups, suchas geodesics in Teichmuller space, to the setting of free groups.Geometric group theory is the study of infinitely symmetricpatterns. Popular examples called ``surface groups'' arefamiliar from wallpaper symmetries and from the symmetries ofEscher's prints. Scientific examples occur in the symmetrygroups of crystalline arrays, and the symmetry groups of fieldtheories in particle physics. The development of topologystarting in the late 19th century, and the concomitantdevelopment of combinatorial group theory, exhibited a directlink between abstract groups and geometry. The need for deeperunderstanding of this link has been demonstrated again and againby different threads within 20th century mathematicaldevelopments. Many of these threads were pulled together around1980 by Gromov, whose proposed unification of geometric grouptheory using the relation of ``quasi-isometry'' has been veryfruitful in the intervening twenty years. The focus of thisresearch project will be to investigate quasi-isometricclassification problems for several different types of symmetrygroups. In particular, by using a constructive technique knownas ``graphs of groups'', new symmetry groups can be constructedout of familiar examples such as surface groups; these andclosely related constructions will be the subjects of thisresearch project.