Hyperbolic 3-space $\mathbb{H}^3$ is the simply connected geodesically complete hyperbolic 3-manifold.
Every hyperbolic 3-manifold is isometric to the quotient space $\mathbb{H}^3/\Gamma$ of hyperbolic 3-space by the action of a torsion-free discrete group acting via isometries.
John Parker, Chapter 5 of Hyperbolic spaces (pdf)
William Abikoff, The bounded model for hyperbolic 3-space and a quaternionic uniformization theorem, Mathematica Scandinavica Vol. 54, No. 1 (August 23, 1984), pp. 5-16 (jstor:24491416)
See also
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