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.
Low dimensional actions of semisimple groups
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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.