Dynamics and the Cohomology of Measured Laminations

In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy.