RUI: Metric and Topological Properties of Self-Similar Groups

主要研究方向为几何群论与拓扑学交叉的两个主要领域。自从Gromov关于多项式增长的著名定理将纯代数性质与群的大规模几何联系起来以来，研究人员一直对通过研究有限生成的无限群的相关凯莱图的几何可以学到什么感兴趣。一项被提议的研究试图将这一项目继续用于某些自相似群体的家庭和这些群体的一些概括。提出的研究的第二个领域源于拓扑不动点理论，并关注每一个自同构具有无限Reidemeister数的群，这一性质具有其他拓扑结果。首席研究员和合作者P. Wong在他们之前的工作的基础上，对这个数字的几何解释感兴趣。主要研究者研究的自相似群和推广包括一类群，其Cayley图（相对于适当的生成集）是Diestel-Leader图，或树的环积，从几何上推广经典的lamplighter群。主要研究者在几何群论领域提出了几个项目。这个领域从几何的角度研究被称为群的基本数学对象。对称群是数学群的一个典型例子。例如，对对称群的研究被用来辨别DNA的形状是双螺旋结构。主要研究者的建议工作与自相似群体有关；从彩色分形到缅因州海岸线的研究，数学和自然界的自相似现象非常普遍。在数学中，自相似群是根据它们在无限树上的作用来定义的；树，或没有电路的图，是研究组合学和计算机科学的基本对象。例如，在计算机科学中，树被用来实现高效的搜索算法。自相似的群体有以特定方式“重新排列”树的指令。上面的研究问题都涉及到这样一个群所附的几何和代数结构之间的相互作用，以及通过学习几何可以学到什么代数性质，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.