Mostow rigidity for skew solenoidal manifolds

We prove a Mostow rigidity theorem for foliated bundles over closed hyperbolic manifolds of dimension $n\ge 3$ endowed with a completely invariant measure of full support. These include solenoidal manifolds obtained as inverse limits of directed systems of finite coverings of closed hyperbolic manifolds. This theorem then extends to skew solenoidal manifolds for which the action of the holonomy group is twisted by means of a cocycle.