Geometric Group Theory

AbstractAward：DMS-0103208首席研究员：Lee Mosher几何群论的激励主题是有限生成群可以使用拓扑和几何方法进行研究。 大约从1980年开始，格罗莫夫提出了利用拟等距关系对群进行几何分类。 他证明了群G的拟等距类通常可以用简单的代数或几何术语明确地描述，这一过程现在被称为群G的“拟等距刚性”或"QI刚性“。本文的第一部分是研究面群图和交换群图的QI-刚性问题。 最初的重点将是那些具有最简单的代数结构的群的图，即与自由群的半直积，由从自由群到适当的自同构群（如曲面的映射类群）的同态决定。 曲面群的一般图是由同态确定到可公度映射类群中的，这是一个更加复杂的课题，这个观点将被用来研究新的有趣的例子的构造。 最近关于阿贝尔群的图的工作也揭示了许多丰富的结构，表明在许多新的情况下获得QI-刚性的真实的可能性。 在这个项目的第二部分，重点将是研究自由群的几何性质，由表面群和自由群之间的类比激发。 特别是，瑟斯顿的结束层压猜想，一个重要的目标，在研究的表面群体，有一个类似的研究freegroups，在分类某些群体行动等变准等距。 追求这一问题将需要推广许多标准的工具，表面集团，如测地线在Teichmuller空间，设置的自由groups.Geometric群论是研究无限dichicpatterns。 被称为“表面基团”的流行例子在墙纸对称性和布里尔印刷品的对称性中很常见。 科学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。 从世纪后期开始的拓扑学的发展，以及伴随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世纪世纪的学术发展中，不同的线索一再证明了对这种联系的深入理解的必要性。 这些线索中的许多线索在1980年左右被格罗莫夫拉到一起，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年里取得了丰硕的成果。 该研究项目的重点是研究几种不同类型对称群的准等距分类问题。 特别是，通过使用一种称为“群的图形”的构造技术，新的对称群可以从熟悉的例子中构造出来，例如曲面群;这些和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.