Cocycle rigidity of partially hyperbolic abelian actions with almost rank-one factors

We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

Smooth cocycle rigidity for expanding maps, and an application to Mostow rigidity

We give a variation on the proof of Mostow's rigidity theorem, for certain hyperbolic 3-manifolds. This is based on a rigidity theorem for conjugacies between piecewise-conformal expanding Markov maps. The conjugacy rigidity theorem is deduced from a Livsic cocycle rigidity theorem that we prove for smooth, compact Lie group-valued cocycles over piecewise smooth expanding Markov maps.