Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

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Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space?

Is there any reference for a proof?