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

jennifer taback首席研究员建议从组合和几何的角度研究各种问题。这些问题中的许多都与汤普森群有关，这是一个令人着迷的群，它出现在数学的各个分支中，从逻辑学到代数同伦论和群论。主要研究人员计划探索从适应性到准等距的问题。此外，研究者建议扩展先前关于格的准等距的工作，与同事合作完成s-算术格的准等距分类。这将完成一个仅由许多几何群理论家完成的项目，该项目引入了有趣的新技术。最后，主要研究者计划了几个关于群的凸性性质的项目。虽然几乎凸性很好理解，但很少有群不是几乎凸但满足较弱凸性条件的例子。她还试图扩大已知几乎是凸的群体的类别。首席研究员提出了群论领域的几个项目。群是一种数学结构，通常在特定对象的对称性背景下引入。一组对称性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，研究者将群体作为几何对象来研究，利用几何来更深入地了解抽象属性。研究者仔细研究了一个特定的群体，以第一个定义这个群体的研究者命名为汤普森群体。汤普森的群体可以用二叉树对从几何上理解，这种方法将这个群体与理论计算机科学中的问题联系起来。因此，它提供了抽象数学的有趣的跨学科应用。此外，汤普森的小组是许多数学问题最喜欢的例子或反例。关于这个群体有许多悬而未决的问题，引起了国际数学家的兴趣。特别是，这个群体的几何形状还没有被很好地理解，这是一个使研究者非常感兴趣的问题。数学家们有一种标准的方式来描述一个群体的“图景”，这种方式有时很难构建。对于某些群体，可以要求计算机构建这幅图。首席研究员感兴趣的是探索一些群体的一种算法性质，称为“几乎凸性”，这使得计算机可以构建群体。

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.