RUI: Metric and Topological Properties of Self-Similar Groups

主要研究者提出建议研究在两个主要领域在于交叉的几何群论和拓扑学。 自从Gromov关于多项式增长的著名定理将一个纯粹的代数性质与群的大规模几何相联系以来，研究人员一直对通过相关Cayley图的几何来研究非线性生成的无限群所能学到的东西感兴趣。 其中一个研究方向是继续研究某些自相似群体的家族以及这些群体的一些推广。 第二个领域提出的研究产生于拓扑不动点理论和关注群体的每一个自同构有无限Reidemeister数，财产有其他拓扑后果。 主要研究者和合作者P. Wong对研究群体的家庭的几何解释感兴趣，建立在他们以前的工作基础上。 自相似的群体和推广的主要研究者包括家庭的群体的凯莱图（关于一个适当的生成集）是Diestel-领导者图，或horocyclic产品的树木，推广几何的经典lampighter groups.The主要研究者提出了几个项目在该地区的几何群论。这个领域从几何的角度研究基本的数学对象，称为组。对称群是数学群的一个典型例子。例如，对称群的研究被用来辨别DNA的形状是双螺旋。首席研究员的拟议工作与自相似群体有关;自相似性在数学和自然中的出现是普遍的，从彩色分形到缅因州海岸线的研究。 在数学中，自相似群是相对于它们在无限树上的作用而定义的;树或没有回路的图是组合学和计算机科学研究的基本对象。 例如，在计算机科学中，树被用来实现有效的搜索算法。 一个自相似的群有指令以一种特殊的方式“重新排列”一棵树。 上述研究问题都涉及到这样一个群体的几何和代数结构之间的相互作用，以及人们可以通过学习几何学来了解代数性质，反之亦然。

The principal investigator proposes proposes research in two main areas which lie at the intersection of geometric group theory and topology. Ever since Gromov's celebrated theorem on polynomial growth which related a purely algebraic property with the large scale geometry of a group, researchers have been interested in what one can learn from studying finitely generated infinite groups via the geometry of their associated Cayley graphs. One line of proposed research seeks to continue this program for certain families of self-similar groups and some generalizations of these groups. The second area of proposed research arises from topological fixed point theory and concerns groups for which every automorphism has infinite Reidemeister number, a property which has other topological consequences. The principal investigator and collaborator P. Wong are interested in a geometric interpretation of this number for the families of groups studied, building on their previous work. The self-similar groups and generalizations studied by the principal investigator include the family of groups whose Cayley graphs (with respect to an appropriate generating set) are Diestel-Leader graphs, or horocyclic products of trees, generalizing geometrically the classical lamplighter groups.The principal investigator proposes several projects in the area of geometric group theory. This field studies fundamental mathematical objects called groups from a geometric point of view. Symmetry groups are a typical example of a mathematical group. The study of symmetry groups was used, for example, to discern that the shape of DNA was a double helix. The proposed work of the principal investigator is related to self-similar groups; the appearance of self-similarity in mathematics and nature is prevalent, from colorful fractals to a study of the Maine coastline. In mathematics, self similar groups are defined relative to their action on an infinite tree; trees, or graphs without circuits, are fundamental objects in the study of combinatorics and in computer science. In computer science, for example, trees are used to implement efficient search algorithms. A self similar group has instructions for a "rearrangement" of a tree in a particular way. The research questions above all relate to the interaction between the geometric and algebraic structure attached to such a group, and what one can learn about algebraic properties by studying geometry, and vice versa.