Quasi-Isometric Rigidity, Convexity Properties and Thompson's Groups

DMS-0305441 Jennifer Taback首席研究员建议从组合和几何的角度研究各种问题。其中许多问题都与汤普森群有关，这是一个迷人的群，存在于从逻辑到代数再到同伦论和群论的不同数学分支中。首席研究员计划探索从适应性到这一组的准等距的各种问题。此外，研究者还建议扩展已有的关于格的拟等距分类的工作，与同事一起完成S算术格的拟等距分类。这将完成一个由许多几何群论家致力于的项目，该项目导致了有趣的新技术的引入。最后，主要研究人员计划了几个关于群的凸性性质的项目。虽然人们对几乎凸性已经有了很好的了解，但很少有群的例子是不几乎凸但满足较弱的凸性条件的。她还试图扩大已知的几乎是凸群的类别列表。主要研究者在群论领域提出了几个项目。群是一种数学结构，通常是在特定对象的对称的背景下引入的。对称集合的某些性质可以外推来描述抽象的数学集合。在她的研究中，研究人员将群体作为几何对象进行研究，使用几何来更好地洞察抽象属性。研究人员仔细研究了一个特殊的群体，以第一个定义它的研究人员的名字命名为汤普森的群体。汤普森的群体可以用二叉树对来几何地理解，这是一种将这个群体与理论计算机科学中的问题联系起来的方法。因此，它提供了一个有趣的抽象数学的跨学科应用。此外，汤普森的小组是数学中许多问题最受欢迎的例子或反例。有许多关于这个群体的悬而未决的问题引起了国际数学家群体的兴趣。特别是，这个群体的几何结构没有被很好地理解，这是一个引起研究者极大兴趣的问题。数学家有一种标准的方法来描述一个群体的“图景”，这有时很难构建。对于某些群体，可以要求计算机来构建这张图片。主要研究人员感兴趣的是探索一些群的一种算法性质，称为几乎最凸性，它允许计算机构造群。

DMS-0305441Jennifer TabackThe principal investigator proposes to study a variety of problemsfrom both a combinatorial and a geometric viewpoint. Many ofthese problems concern Thompson's group, a fascinating group whichappears in varied branches of mathematics, from logic to algebrato homotopy theory and group theory. The principal investigatorplans to explore questions ranging from amenability toquasi-isometries of this group. Additionally, the investigatorproposes to extend prior work on quasi-isometries of lattices,working with colleagues to finish a quasi-isometry classificationof s-arithmetic lattices. This would complete a project worked onby many geometric group theorists that has led to the introductionof interesting new techniques. Finally, the principal investigatorplans several projects concerning convexity properties of groups.While almost convexity is well understood, there are few examplesof groups which are not almost convex but satisfy weaker convexityconditions. She also seeks to expand the list of classes ofgroups which are known to be almost convex.The principal investigator proposes several projects in the areaof group theory. A group is a mathematical structure, oftenintroduced in the context of symmetries of a particular object.Certain properties of a set of symmetries can be extrapolated todescribe abstract mathematical sets. In her research, theinvestigator studies groups as geometric objects, using thegeometry to give greater insight into abstract properties. Theinvestigator studies closely one particular group, namedThompson's group after the researcher who first defined it.Thompson's group can be understood geometrically using pairs ofbinary trees, an approach which relates this group to questions intheoretical computer science. Thus it provides an interestinginterdisciplinary application of abstract mathematics.Additionally, Thompson's group is a favorite example orcounter-example to many questions in mathematics. There are manyopen questions about this group which interest an internationalgroup of mathematicians. In particular, the geometry of thisgroup is not well understood, a question which interests theinvestigator greatly. Mathematicians have a standard way ofdescribing a "picture" of a group, which is sometimes difficult toconstruct. For certain groups, it is possible to ask a computer toconstruct this picture. The principal investigator is interestedin exploring an algorithmic property of some groups, called almostconvexity, which allows computer construction of the group.