We know by Thurston's Geometrization Conjecture, that every closed 3-manifold admits a prime decomposition: it must be the connected sum of prime 3-manifolds.
My question is: if $\mathbb{X}$ is one of Thurston's 8 geometrical models, and $\Gamma < \text{Diffeo}(\mathbb{X})$ is a cocompact subgroup (not necessarily a subgroup of isometries) then the quotient $M=\mathbb{X}/\Gamma$ is a prime 3-manifold? Or it could admit a non-trivial decomposition (in the sense that it could admit more than one prime manifold glued together)?
Thanks in advance for any help!
Edit: To be clear, what I truly want to know is if it is possible for a quotient $M=\mathbb{X}/\Gamma$ by a cocompact subgroup $\Gamma < \text{Diffeo}(\mathbb{X})$, that acts freely and discrete on $\mathbb{X}$, to have distinct regions locally isometric to distinct model geometries.