Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism f on a closed 3-manifold with exponential growth in its fundamental group such that f n is not isotopic to the identity for all n 0. This example contradicts a conjecture in [HHU1]. The main idea is to consider a well-understood time-t map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.