Discrete subgroups of semisimple Lie groups

University Of California-San Diego, La Jolla CA

Investigators

Abstract

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.