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Take $\tilde{\Gamma}$ to be a higher genus Baumslag-Solitar group, see section 9 of M. Kapovich, B. Kleiner, Coarse Alexander duality and duality groups, Journal of Differential Geometry, Vol. 69, (2005) Number 2, p. 279-352. Group $\tilde\Gamma$ cannot act properly discontinuously on any contractible 3-manifold, but it contains a finite index subgroup $\Gamma$ which is is Gromov-hyperbolic and isomorphic to the fundamental group of a compact 3-manifold with boundary. Hence, by a corollary of Thurston's geometrization theorem, $\Gamma$ acts isometrically and discretely on hyperbolic 3-space, i.e., is a discrete subgroup of $G=PO(3,1)$.