Growth of lattices in semisimple Lie groups

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.