Counting Manifolds and Embeddings of Free Groups

拟议的研究包括两个主要项目。第一个，计算流形，在某种意义上是渐近群论的连续类比。其指导思想是将局部对称流形的有限性结果从刚性现象和算术性转化为具体的定量表述。它与数学中的一些核心问题，如同余子群问题，以及Borel和Prasad著名的有限定理密切相关，在黎曼几何、数论和理论物理中都有应用。该项目延续了私家侦探的早期工作，以及私家侦探与Burger、Lubotzky和Mozes的联合工作。第二个主要项目是将自由群划分成具有某种几何结构的群。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，并影响着微分几何、遍历理论、几何群论、酉表示和无限群的一些主题。私家侦探与布鲁拉德（E. Breuillard）合作追求的目标之一，是找到Tits替代方案的有效版本，该方案的弱版本已被埃斯金（Eskin）、莫泽斯（Mozes）和欧（Oh）在解决格罗莫夫（Gromov）的指数增长猜想时证明。其他问题与澳大利亚人猜想有关。本课题还涉及解析李群的密集子群的研究，以及对给定可数群的（解析）度量补全分类的“相反”问题。关于局部对称空间，有几个经典的有限陈述已经被发现了30多年，但没有定量的证明，或者现有的估计是次优的。其中一个例子是Wang关于有界体积流形数量有限的经典定理（以及由Borel和Prasad给出的强版本）；我们希望对这个数字有一个准确的估计。另一个例子是有限体积的流形的基本群是有限呈现的；最小演示文稿的大小可以根据体积来估计。更一般地说，我们研究流形的体积与其几何结构之间的关系。第二个项目处理自由子组。在他1972年著名的论文J. Tits中，证明了任何有限生成的不可虚解的线性群都包含一个非交换的自由子群。这个结果，今天被称为Tits替代，回答了Bassand Serre的一个猜想，是理解线性群的重要一步。对提茨定理的任何改进都会在数学的各个不同领域产生直接的推论。P.I.和e.b oulillard最近建立了Tits定理的拓扑版本，它回答了动力学、黎曼叶化和无限群中的几个问题。

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.