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。 这些问题中的许多涉及汤普森集团，一个迷人的集团，出现在不同的数学分支，从逻辑代数同伦理论和群论。 主要作者计划探讨这个群体的问题，从顺从性到准等距性。 此外，作者还提出了扩展先前关于格的拟等距的工作，与同事们一起完成了s-算术格的拟等距分类。 这将完成一个由许多几何群理论家工作的项目，该项目导致了有趣的新技术的引入。最后，主要作者提出了几个关于群的凸性的方案，虽然几乎凸性已经被很好地理解，但很少有不几乎凸但满足较弱凸性条件的群的例子。 她还试图扩大名单类ofgroups这是众所周知的几乎凸。首席研究员提出了几个项目在areaofgroup理论。群是一种数学结构，通常在特定对象的对称性的上下文中引入。对称性集合的某些属性可以外推来描述抽象的数学集合。 在她的研究中，研究者将群体作为几何对象进行研究，利用几何学来更深入地了解抽象属性。研究人员仔细研究了一个特殊的组，命名为汤普森组后，研究人员谁第一次定义它。汤普森的组可以理解几何使用对二叉树，一种方法，涉及这个组的问题，在理论计算机科学。因此，它提供了一个有趣的跨学科应用的抽象数学。此外，汤普森的小组是一个最喜欢的例子或反例，许多问题的数学。关于这个群体有许多悬而未决的问题，引起了一个国际数学家团体的兴趣。 特别是，这个群的几何形状还没有很好地理解，这是一个令研究者非常感兴趣的问题。 数学家们有一个标准的方法来描述一个群体的“图像”，这有时很难构建。对于某些群体，可以让计算机来构建这幅图。 主要研究者对探索一些群的算法性质感兴趣，称为几乎凸性，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.