Smooth Cocycles over Homogeneous Dynamical Systems.

In this thesis, we study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra---one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows that come from an embedded copy of $overline{SL(2,RR)}^{l_{1}} times overline{SL(2,RR)}^{l_{2}}$. In both cases we show that any smooth $RR$-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function. (This is commonly referred to as smooth cocycle rigidity.) These build on results of D. Mieczkowski, where the same is shown for actions on $(SL(2,RR) times SL(2,RR))/G$. These results yield a number of corollaries, particularly for group actions that restrict to either of the two types of aforementioned actions. For example, we prove smooth cocycle rigidity for the action of the upper-triangular (and strictly upper-triangular) subgroup of $SL(n,RR)$ on $SL(n,RR)/G$, for $n geq 3$ (respectively, $n > 3$). We also prove smooth cocycle rigidity for higher-rank abelian actions that restrict to one of the two above mentioned actions. As an application, we use this last result to establish rigidity for smooth time-changes of these higher-rank abelian actions.Ph.D.MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77709/1/felipear_1.pd