Counting Manifolds and Embeddings of Free Groups

计算自由群的流形和嵌入

• 批准号: 0404557
• 负责人: Tsachik Gelander
• 金额: $ 10.5万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-12-15 至 2007-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0404557&HistoricalAwards=false
• 关键词: Counting; Manifolds; Embeddings; Free; Groups

The proposed research includes two main projects. The first one, countingmanifolds, is in a sense a continuous analog of asymptotic group theory. The guiding line is to convert finiteness results about locally symmetricmanifolds, which follow from rigidity phenomenon and arithmeticity, toconcrete quantitative statements. It is closely related to some centralquestions in mathematics such as the congruence subgroup problem, and tothe remarkable finiteness theorem of Borel and Prasad, and it hasapplications in Riemannian geometry, number theory and theoreticalphysics. This project continues earlier work of the P.I. and joint work ofthe P.I. with Burger, Lubotzky and Mozes. The second main project concernsembeddings of free groups into groups with some geometric structure. Thisplays a central role in the study of linear and topological groups (inparticular subgroups of Lie groups over local fields), and impacts sometopics in differential geometry, ergodic theory, geometric group theory,unitary representations and profinite groups. One target, which the P.I.pursues in collaboration with E. Breuillard, is to obtain an effectiveversion to Tits alternative, a weak version of which was proved by Eskin,Mozes and Oh, while solving Gromov's exponential growth conjecture. Otherproblems are related to the Auslander conjecture. This project is alsorelated to the study of dense subgroups of analytic Lie groups, and the``opposite'' problem of classifying the (analytic) metric completions of agiven countable group.There are several classical finiteness statement concerning locallysymmetric spaces which have been known for more than 30 years, and yethave no quantitative proofs, or for which the existing estimates aresuboptimal. One example is the classical theorem of Wang (and its strongversion due to Borel and Prasad) about the finiteness of the number ofmanifolds with bounded volume; we would like to have good estimates forthis number. Another example is the fact that the fundamental group of amanifold with finite volume is finitely presented; the size of a minimalpresentation can be estimated in terms of the volume. More generally, westudy relations between the volume of manifolds and their geometricstructure. The second project deals with free subgroups. In his celebrated1972 paper J. Tits proved that any finitely generated linear group whichis not virtually solvable contains a non-commutative free subgroup. Thisresult, known today as the Tits alternative, answered a conjecture of Bassand Serre and was an important step toward the understanding of lineargroups. Any improvement in Tits' theorem has immediate corollaries invarious different fields of mathematics. The P.I. and E. Breuillard hadrecently established a topological version of Tits theorem which answeredseveral questions in dynamics, Riemannian foliations and profinite groups.

拟议的研究包括两个主要项目。第一个，countingmanifold，在某种意义上是渐近群理论的连续模拟。 其指导思想是把由刚性现象和算术性所得出的局部扩张流形的有限性结果转化为具体的定量陈述。它与一些数学中心问题如同余子群问题、Borel和Prasad的显着有限性定理等密切相关，并在黎曼几何、数论和理论物理中有应用。该项目延续了P.I.的早期工作。和私家侦探的合作和汉堡，卢博茨基和莫兹一起第二个主要项目concernsembeddings的自由团体成团体与一些几何结构。这在线性和拓扑群（特别是局部域上李群的子群）的研究中起着核心作用，并影响了微分几何，遍历理论，几何群论，酉表示和profinite群中的一些主题。 一个目标，其中PI追求与E。Breuillard提出的一种新的指数增长猜想是在解决Gromov的指数增长猜想时，得到了Eskin，Mozes和Oh证明的Tits替代方案的一个有效版本。这类问题与Auslander猜想有关。这个项目也涉及到解析李群的稠密子群的研究，以及可数群的度量完备化的分类问题，有几个经典的关于局部对称空间的有限性陈述已经知道了30多年，但没有定量的证明，或者现有的估计是次优的。一个例子是王的经典定理（以及博雷尔和普拉萨德的强版本）关于有界体积的流形数量的有限性;我们希望对这个数字有很好的估计。另一个例子是有限体积的流形的基本群是极小表示的，极小表示的大小可以用体积来估计。更一般地说，我们研究流形的体积和它们的几何结构之间的关系。第二个项目涉及自由子群。在他著名的1972年的论文J. Tits证明，任何生成的线性群whichis不是虚拟可解包含一个非交换的自由子群。这一结果，今天被称为山雀替代，回答了猜想的Bassand塞尔，是一个重要的一步了解lineargroups。提茨定理的任何改进在数学的各个不同领域都有直接的推论。私家侦探和E. Breuillard最近建立了一个拓扑版本的Tits定理，它回答了动力学，黎曼叶理和profinite群中的几个问题。