Relative spectrum for smooth actions of Kazhdan groups on fiber bundles

Abstract

In this paper we indicate how techniques developed by Zimmer [8] regarding certain smooth actions of discrete groups on compact manifolds can be extended to actions of discrete groups and semisimple Lie groups on compact fiber bundles. The actions that we consider preserve a smooth volume density on the total space of the bundle that defines a smooth field of volume densities on the fibers. When the standard fiber has ‘low’ dimension or the action preserves certain geometric structures on the fibers, Zimmer's techniques can show that the action must be measurably conjugate to a topologically relatively isometric action.