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Do there exist compact Riemannian manifolds of constant negative sectional curvature in all dimensions $\geq2$?

The two-dimensional case is well-known. For higher dimensions, this question asks for explicit examples, and it turns out that the construction could be very difficult. But what if I only ask for existence? Is there an easy way to show that that $\mathbb{H}^n$ admits a compact quotient manifold for all $n\geq2$?

Thanks in advance!

Yuxiao Xie
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    It is a theorem of Borel that all linear real semisimple Lie groups have a cocompact Lattice, you can check that the isometries of hyperbolic space satisfy this. – Nick L Apr 06 '20 at 07:47
  • @NickL Is the quotient necessarily a manifold then? – Yuxiao Xie Apr 06 '20 at 11:38
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    @Colescu: Selberg proved that every finitely generated matrix group contains a torsion-free subgroup of finite index. Hence, after passing to a finite index subgroup, the quotient will be a manifold in your setting. – Moishe Kohan Apr 06 '20 at 16:29

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