Trees and Group Actions

CARBONE，DMS 98- 00604提出者正在研究局部有限树的自同构群，这是局部紧群，以及它们的有限余体积的离散子群，这被称为树格。 这个提议者最近发现了如何构造非均匀树格（及其子格）;这些是由海曼·巴斯和亚历克斯·卢博茨基证明存在的。 特别令人感兴趣的是树是两个可能度的齐次树的情况，每个度都是素数加1的幂。 对于一定程度的双齐性，树则是非阿基米德局部域上的秩1单李群的Bruhat-Tits构造。 广义地说，我们感兴趣的是构造包含在这种秩1李群中的非均匀格，将它们与一般树格进行比较，进一步研究非均匀格的存在性问题，它们的余体积和算术性问题。提出者还研究了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”是没有闭合回路的连通图，以及它们的对称性的代数结构。 提出者最近发现，存在某些局部有限的树的对称性，称为“非均匀树格”，这允许我们将无限树折叠成在适当意义上具有有限体积的无限“商图”。 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，并且还知道这些会产生无限族的有限扩展图，这是计算机科学中某些通信网络的基本构建块。 非均匀树格及其商图的构造，以及从许多数学观点来看是自然的，是期望的，以便在通信网络的构造中可能有新的实际应用。

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.