Abstract
We prove that SL (n, ℚ) has no nontrivial, C∞, volume-preserving action on any compact manifold of dimension strictly less than n. More generally, suppose G is a connected, isotropic, almost-simple algebraic group over ℚ, such that the simple factors of every localization of G have rank ≥ 2. If there does not exist a nontrivial homomorphism from G(ℝ)° to GL (d, ℂ), then every C∞, volume-preserving action of G(ℚ) on any compact d-dimensional manifold must factor through a finite group.
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