Mathematical Sciences: Negatively Curved Groups and Tilings

数学科学：负曲群和平铺

基本信息

批准号：
9102471
负责人：
Mark Feighn
金额：
$ 4.1万
依托单位：
Rutgers University New Brunswick
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1991
资助国家：
美国
起止时间：
1991-07-01 至 1994-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9102471&HistoricalAwards=false
关键词：
Mathematical Sciences Negatively Curved Groups

Mathematical Sciences Negatively Curved Groups

项目摘要

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.

最简单的无限组是有限数量字母的免费组。在这里，小组的一个元素是字母及其倒置中的一个单词。且仅当迭代取消连续发生字母的发生及其倒数时，两个单词在两个单词变为相同时，两个单词是平等的。格罗莫夫（Gromov）启动了一类称为“负弯曲”的组类别的研究，这些群体具有自由组的一些计算易度性。该项目的第一部分是继续探索这些群体，并将这项研究扩展到包括研究者称为“相对负弯曲”的一类小组。我们说，如果我们将平面表示为有限的一致零件，瓷砖，它们是成对的，直到其边界都没有相交的瓷砖，我们已经铺平了平面。瓷砖不必连接，其边界可能是分形的。如果存在与瓷砖相关的复杂数字B，则瓷砖是自相似的，以使B的乘法将每个瓷砖带到瓷砖的结合。一个例子就是著名的彭罗斯瓷砖。瑟斯顿（Thurston）已经确定了可以与自相似瓷砖相关的复数。该项目的第二部分是确定哪些复数可能与磁盘的瓷砖有关，哪些数字可能与多边形有关。