This proposal addresses problems on discrete subgroups of semisimple Lie groups. These problems have been the interplay of several kinds of mathematics. They are, on one hand, related to counting curves over a finite field and, on the other, to the asymptotic behavior of the subgroup growth of lattices. The PI has already solved several problems in this area and would like to give a better picture. The second purpose of this proposal is to describe the "smallest" locally symmetric orbifolds with a given covering space, or its equivalent object in the non-Archimedean setting. This natural question had been considered by many mathematicians such as Siegel, Chinburg, Friedman, Meyerhoff, Gehring, Martin, and Lubotzky, and it is solved for lattices in SL(2) over different local fields. The PI solved this problem for most of Chevalley groups over a positive characteristic local field. The PI would like to understand structure of such orbifolds. For instance, he plans to answer Lubotzky?s question on whether the smallest orbifold is compact. The next problem is the classification of discrete vertex transitive actions on Bruhat-Tits buildings. These actions have been of interest since the 80?s. Several mathematicians have tried to construct such actions and, so far, for large dimensions, only one family of such actions have been constructed. They have been also used to construct explicit Ramanujan complexes. These combinatorial objects are generalization of Ramanujan graphs, which are highly useful in computer science, and expected to have broad applications. The PI plans to classify such actions. The PI and Mohammadi have constructed new families of simply-transitive actions on the vertex set of the Bruhat-Tits building and gave a very strong classification theorem, and it might be possible to get new examples in positive characteristic which in turn would give us new Ramanujan complexes. The PI proposes a step toward a recent conjecture by Sarnak on equi-distribution of orbits of prime powers of a unipotent element in a finite volume homogeneous space.
Discrete subgroups of semi-simple groups can be considered as the connecting point of several parts of mathematics. On one hand, they are related to geometry and geometric group theory, and on the other to dynamical systems and number theory. The PI proposes several related problems on this subject. In these projects, PI would like to either count number of certain combinatorial objects with rich algebraic structure, or describe "smallest" models with certain descriptions, which usually have more symmetries and might be useful in computer science, or seek for other evidences of the randomness of the Möbius function. These projects consist of different parts and of various mathematical nature, e.g. Bruhat-Tits theory, mass formula, subgroup growth and sieve theory. Because of its relations with a wide range of mathematics, students at both graduate and under-graduate level can be exposed to and learn different topics. Moreover, some parts are of combinatorial or computational nature which makes them more accessible to under-graduates.
Often symmetries of an object tell us a lot about it. This methodology has been used in physics to understand the geometry of universe and in chemistry to understand the structure of a crystal. For these reasons, among other things, at the end of the nineteenth century, mathematicians launched group theory to study the structure of symmetries themselves. A good part of math and sciences is about finding patterns to simplify a complex structure. And group theory is about the rules of patterns. Two of the easiest patterns are tilling a line with a segment of length one and a plane with a one-by-one square. The patterns can be described using the integer translations. So in the first case the (additive group of) integers and in the second case the 2D integer vectors give us the pattern. Using results from various parts of math, the PI describes smallest volume tiles of a space called Bruhat-Tits building. Interesting enough, it is showed that these minimum volume tiles are not bounded. They are showed to be unique in certain cases. The PI and his coauthors give the precise description of spaces whose vertices can be covered by (discrete) translations of a single vertex. These kind of spaces had been used to construct the so called Ramanujan complexes. These are highly connected sparse complexes, and the general belief is that these complexes are useful in error-correcting codes. As another part of this project, the PI and his coauthor give a robust way of constructing highly connected sparse graphs . Such graphs are extremely important in communication and computer science. And in the past decade they have been found to be extremely useful in various parts of math, e.g. number theory, group theory, and topology. The PI designed several graduate courses on the related topics and made them accessible to first year graduate students. The PI coorganized several workshops and seminars, and shared his results with the community either by participating in several workshops and seminars or by making his works available to the public
