Geometric Group Theory

[摘要]获奖：dms - 0103308首席研究员：Lee mosher几何群论的激励主题是可以使用拓扑和几何方法研究有限生成群。从1980年左右开始，Gromov提出使用准等距关系从几何上对群进行分类。他证明了G群的拟等距类通常可以用简单的代数或几何术语来明确描述，这个过程现在被称为G群的“拟等距刚性”或“qi -刚性”。这个项目的第一部分将是研究曲面群和阿贝群图的qi -刚性问题。最初的重点将是那些具有最简单代数结构的群的图，即与自由群的半直接积，由自由群到适当自同构群（如曲面的映射类群）的同同态决定。面群的广义图是由同态映射到可测映射类群（一个更神秘的对象）来决定的，这个观点将用于研究新的和有趣的例子的构造。此外，最近关于阿贝尔群的工作图揭示了许多丰富的结构，这表明在许多新情况下获得qi -刚性的真正可能性。在这个项目的第二部分，重点将是研究自由群的几何性质，由表面群和自由群之间的类比驱动。特别地，Thurston的结束层叠猜想是研究表面群的一个重要目标，在自由群的研究中有类似的情况，即将某些群作用分类到等变拟等长。为了解决这个问题，需要将曲面群的许多标准工具，如Teichmuller空间中的测地线推广到自由群的设置中。几何群论是对无限对称图案的研究。流行的例子被称为“表面群”，从墙纸对称和埃舍尔版画的对称中我们都很熟悉。科学上的例子出现在晶体阵列的对称群和粒子物理学中场论的对称群中。拓扑学的发展始于19世纪后期，与此同时，组合群论的发展显示了抽象群与几何之间的直接联系。在20世纪的数学发展中，对这种联系进行更深入理解的必要性已经被不同的线索一次又一次地证明了。1980年前后，Gromov把这些线索集中在一起，他提出了用“准等距”关系统一几何群论的建议，在这20年里取得了丰硕的成果。本研究的重点是研究几种不同类型对称群的准等距分类问题。特别地，通过使用一种被称为“群图”的构造技术，新的对称群可以从熟悉的例子中构造出来，比如曲面群；这些和密切相关的结构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.