Abstract:
A simple example of an arithmetic group, but one which will be sufficiently interesting for the purposes of the talk, is the group SL(n,Z) of determinant one integral matrices. We shall discuss four properties of arithmetic groups: The congruence subgroup property, superrigidity, property (T) of Kazhdan, and bounded generation. Each of these properties involves deep tools from different areas of mathematics: algebra, geometry, ergodic theory, number theory, group theory and infinite dimensional representation theory. A phenomenon which is currently partially explained and partially conjectured, is that these properties appear together. In the talk we shall carefully explain all these properties, the connections and conjectures involved, and some of the recent developments. A background of undergraduate studies will be more than enough.

Coffee, Cookies, Company at the faculty lounge at 3:30.