ON THE SPECTRAL THEORY OF GROUPS OF AUTOMORPHISMS OF S-ADIC NILMANIFOLDS

Let X be an S-adic compact nilmanifold, equipped with its unique translation invariant probability measure µ. We characterize the countable groups Γ of automorphisms of X for which the Koopman representation κ on L 2 (X, µ) has a spectral gap. More specifically, we show that κ does not have a spectral gap if and only if there exists a non-trivial Γ-invariant quotient solenoid (that is, a finite dimensional, connected, compact abelian group) on which Γ acts as a virtually abelian group