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Published online by Cambridge University Press: 13 May 2022
In this paper we study Zimmer's conjecture for $C^{1}$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of an uniform lattice is larger than the dimension of the manifold, then the action factors through a finite group. For lattices in ${\rm SL}(n, {{\mathbb {R}}})$, the dimensional bound is sharp.
Brown was supported by NSF No.1752675. Damjanović was supported by Swedish Research Council grant VR2015-04644. Zhang was supported by the National Science Foundation under Grant No. DMS-1638352.