I have been studying symmetric spaces, locally symmetric spaces, arithmetic groups and related discrete groups from points of view of geometry, topology and analysis. SL(n, Z) is one of the most basic examples of arithmetic groups, and it acts on the symmetric space of positive definite matrices of determinant 1. The quotient of the symmetric space by SL(n, Z) is a locally symmetric space, which occurs naturally in many subjects from differential geometry, topology, geometry group theory to number theory. Though these spaces and groups have been studied extensively by many people, there are still many open problems.
I have also been studying two closely related transformation groups:
(1) Mapping class groups and their action on the Teicmuller spaces of marked Riemann surfaces
(2) Outer automorphism groups of free groups Out(F_n) and their action on the outer spaces of marked metric graphs.
The close analogy of these two actions and the action of arithmetic groups on symmetric spaces has motivated a lot of work. They are probably three of the most important classes of groups in geometric group theory. Many new problems are waiting to be understood and solved.
