Invariant Cocycles, Random Tilings and the Super-K and Strong Markov Properties

We consider 1-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many of the results in [9] to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.