Trees and Group Actions

树木和群体行动

• 批准号: 9800604
• 负责人: Lisa Carbone
• 金额: $ 8.48万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800604&HistoricalAwards=false
• 关键词: Trees; Group; Actions

CARBONE, DMS 98-00604The proposer is studying automorphism groups of locally finite trees, which are locally compact groups, and their discrete subgroups of finite covolume, which are called tree lattices. The proposer has recently discovered how to construct non-uniform tree lattices (and their quotients); these were conjectured by Hyman Bass and Alex Lubotzky to exist. Of particular interest is the case where the tree is homogeneous of two possible degrees,each a power of a prime plus one. For certain degrees of bi-homogeneity, the tree is then the Bruhat-Tits building of a rank 1 simple Lie group over anon-archimedean local field. Broadly speaking, we are interested in constructingnon-uniform lattices contained within such rank 1 Lie groups, comparing them with general tree lattices, further questions of existence of non-uniform lattices, their covolumes and questions of arithmeticity. The proposer is also studying the edge-indexed quotient graphs of non-uniform tree lattices, which give new examples of infinite expanding diagrams.This is a study of infinite 'trees', which are connected graphs in which there are no closed circuits, and the algebraic structure of their symmetries. The proposer has recently discovered that there exist certain symmetries of trees which are locally finite, called 'non-uniform tree lattices', that allow us to collapse the infinite tree into an infinite 'quotient graph' that has finite volume in an appropriate sense. The symmetries of locally finite trees that give finite quotient graphs have been understood for some time, and it is also known that these give rise to infinite families of finite expander graphs, which are fundamental building blocks for certain communication networks in computer science. The construction of non-uniform tree lattices and their quotient graphs, as well as being natural from many mathematical points of view, is desirable in order that there may be new practical applications in the construction of communication networks.

CARBONE，DMS 98-00604提议者正在研究局部有限树的自同构群（局部紧群）及其有限余体积的离散子群（称为树格）。 提议者最近发现了如何构造非均匀树格（及其商）；海曼·巴斯和亚历克斯·卢博茨基推测这些是存在的。 特别令人感兴趣的是树具有两个可能的度的齐次性的情况，每个度都是素数加一的幂。 对于一定程度的双同质性，该树是非阿基米德局部域上的 1 阶简单李群的 Bruhat-Tits 构建。 从广义上讲，我们感兴趣的是构造包含在此类 1 阶李群中的非均匀格，将它们与一般树格进行比较，进一步讨论非均匀格的存在性、它们的余体积和算术问题。提议者还在研究非均匀树格的边索引商图，它给出了无限扩展图的新例子。这是对无限“树”的研究，无限“树”是没有闭合电路的连通图，以及它们对称性的代数结构。 提议者最近发现，存在某些局部有限的树对称性，称为“非均匀树格”，它允许我们将无限树折叠成一个无限的“商图”，在适当的意义上具有有限的体积。 给出有限商图的局部有限树的对称性已经被理解了一段时间，而且众所周知，它们产生了无限族的有限扩展图，它们是计算机科学中某些通信网络的基本构建块。 为了在通信网络的构建中可能有新的实际应用，非均匀树格及其商图的构造，以及从许多数学角度来看是自然的，是可取的。