Growth of lattices in semisimple Lie groups

H.C.Wang和A.Borel的一个经典定理说，在给定的单李群中，有界余体积的算术格的数目是有限的，直到自然等价。我们感兴趣的问题是，有多少算术格的余体积有界于某个给定的x？晶格的数量相对于x渐近增长的速度有多快？关于群上的某个Haar测度，计算了格子的余体积。一个说明性的例子是当Haar度量被选为Serre意义下的Euler-Poincare度量时。在这种情况下，余体积就是相应因子空间的欧拉特征，它是空间复杂性的度量。任意Haar度量的一般情况被视为类似于此。因此，我们感兴趣的是研究具有固定下层结构的空间的复杂性的定量性质。使问题特别丰富和引人入胜的是算术味道。一般而言，著名的Marguis定理提供了通向算术性的桥梁，该定理指出，在实数阶的李群中，所有格都是算术定义的。这是超刚性原理的结果，该原理也是G.A.马古利斯发现的。因此，我们感兴趣的问题位于代数、数论和几何之间的某个边界上。即使是对其中一些问题的部分回答，也可能对这些主题和相关主题产生不平凡的影响。事实上，我们已经有了几个这种影响的显著例子，还有一些其他的正在调查中。对应用程序的研究是该项目的重要组成部分。使这个项目适时的是各种可用的方法和结果，其中一些是最近的，这为建议的研究提供了坚实的背景。

A classical theorem of H. C. Wang and A. Borel says that in a given simple Lie group the number of arithmetic lattices of bounded covolume is finite up to a natural equivalence. The questions we are interested in are How many arithmetic lattices are there with covolumes bounded by some given x? and How fast the number of lattices grows asymptotically with respect to x? The covolume of a lattice is calculated with respect to some Haar measure on the group. An illustrative example is when the Haar measure is chosen to be the Euler-Poincare measure in the sense of Serre. In this case the covolume is just the Euler characteristic of the corresponding factor space, it is a measure of complexity of the space. The general case of an arbitrary Haar measure is seen to be similar to this. Therefore we are interested in studying quantitative properties of the complexity of spaces with some fixed underlying structure.What makes the questions particularly rich and engaging is the arithmetic flavour. In general the bridge to arithmeticity is provided by the celebrated Margulis theorem which states that in Lie groups of real rank at least 2 all lattices are defined arithmetically. This is a consequence of the superrigidity principle which was also discovered by G. A. Margulis.Thus the questions we are interested in sit somewhere on the border between algebra, number theory and geometry. Even partial answers to some of them may have non-trivial consequences in these and related subjects. In fact, we already have several striking examples of such implications and some other are being investigated. The study of applications is an essential part of the project. What makes this project timely is the variety of available methods and results, some being very recent, which provide a solid background for a suggested research.