Mathematical Sciences: Negatively Curved Groups and Tilings

最简单的无限群是有限字母上的自由群。在这里，组中的一个元素是字母及其反义词中的一个单词。两个单词相等当且仅当在迭代地取消连续出现的字母及其逆之后，两个单词变得相同。格罗莫夫开始研究一类群，称为“负弯曲”，它们具有自由群的一些计算便利性。该项目的第一部分是继续探索这些群体，并将研究扩展到研究者称之为“相对负曲线”的一类群体。如果我们将复平面表示为有限多个全等块的并集，我们就说我们已经平铺了复平面，这些平铺块，在它们的边界上是两两不相交的。瓷砖不需要连接，它们的边界可能是分形的。如果有一个与平铺相关的复数b，则平铺是自相似的，因此乘以b将每个平铺变为一个平铺的并集。著名的彭罗斯瓷砖就是一个例子。瑟斯顿已经确定了可以与自相似平铺相关联的复数。该项目的第二部分是确定哪些复数可能与磁盘的平铺相关联，哪些数字可能与多边形的平铺相关联。

The simplest of infinite groups is the free group on a finite number of letters. Here an element of the group is a word in the letters and their inverses. Two words are equal if and only if after iteratively canceling consecutive occurrences of a letter and its inverse the two words become the same. Gromov initiated the study of a class of groups, called "negatively curved," that share some of the computational ease of a free group. The first part of this project is to continue the exploration of these groups and to extend this study to include a class of groups that the investigator calls "relatively negatively curved." We say we have tiled the complex plane if we have expressed the plane as a union of finitely many congruent pieces, the tiles, that are pairwise disjoint up to their boundaries. The tiles need not be connected and their boundaries may be fractal. The tiling is self-similar if there is a complex number b associated to the tiling such that multiplication by b takes each tile to a union of tiles. An example is the famous Penrose tiling. Thurston has identified the complex numbers that can be associated to self-similar tilings. The second part of this project is to determine which complex numbers may be associated to tilings by disks and which numbers may be associated to tilings by polygons.