Kazhdan's Property T for the Symplectic Group over a Ring

Abstract

Kazhdan constants for $\mathop{\rm {Sp}}\left( n,\Omega \right) $ where $\Omega $ is a commutative topological ring with dense finitely generated subring with unity are determined. This implies Kazhdan's property T for these groups. As application explicit Kazhdan constants are determined for the loop groups corresponding to $\mathop{\rm {Sp}}\left( n,\mathbf{C}\right) $. These are further examples of groups with property T which are infinite dimensional Lie groups and not locally compact.