Lattices in simple p-adic groups have only finite C1 actions on compact manifolds of dimension less than the rank; extended to totally disconnected S-arithmetic groups with critical dimension equal to maximal rank of simple factors.
Recent progress in the Zimmer program
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abstract
This paper can be viewed as a sequel to the author's long survey on the Zimmer program \cite{F11} published in 2011. The sequel focuses on recent rapid progress on certain aspects of the program particularly concerning rigidity of Anosov actions and Zimmer's conjecture that there are no actions in low dimensions. Some emphasis is put on the surprising connections between these two different sets of developments and also on the key connections and ideas for future research that arise from these works taken together.
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Actions of lattices in $S$-arithmetic groups on manifolds
Lattices in simple p-adic groups have only finite C1 actions on compact manifolds of dimension less than the rank; extended to totally disconnected S-arithmetic groups with critical dimension equal to maximal rank of simple factors.