Geometrization conjecture

A $3$-manifold is closed if it is compact and has empty boundary. A closed $3$-manifold $M$ is prime if whenever $M$ is homeomorphic to a connected sum $M_1 \# M_2$, one of $M_1, M_2$ is homeomorphic to $S^3$.

A model geometry is a pair $(X,G)$ consisting of a simply connected smooth manifold $X$ together with a smooth transitive action of a Lie group $G$ on $X$ whose point stabilizers are compact; it is maximal if $G$ is maximal among Lie groups acting smoothly and transitively on $X$ with compact stabilizers. A geometric structure on a manifold $N$ modelled on $(X,G)$ is a diffeomorphism from $N$ to a quotient $X/\Gamma$, where $\Gamma \le G$ is a discrete subgroup acting freely on $X$. Fixing a $G$-invariant Riemannian metric on $X$, such a structure has finite volume if the induced metric on $N$ gives $N$ finite total volume.

Up to equivalence there are exactly eight maximal $3$-dimensional model geometries admitting at least one compact quotient manifold, namely those with $X$ equal to

$$S^3,\quad \mathbb{E}^3,\quad \mathbb{H}^3,\quad S^2 \times \mathbf{R},\quad \mathbb{H}^2 \times \mathbf{R},\quad \widetilde{\mathrm{SL}}(2,\mathbf{R}),\quad \mathrm{Nil},\quad \mathrm{Sol},$$

with $G$ in each case the full group of isometries of the corresponding invariant metric: $O(4,\mathbf{R})$ for $S^3$; $\mathbf{R}^3 \rtimes O(3,\mathbf{R})$ for $\mathbb{E}^3$; $O^{+}(1,3,\mathbf{R})$ for $\mathbb{H}^3$; $O(3,\mathbf{R}) \times \mathbf{R} \times \mathbf{Z}/2\mathbf{Z}$ for $S^2 \times \mathbf{R}$; $O^{+}(1,2,\mathbf{R}) \times \mathbf{R} \times \mathbf{Z}/2\mathbf{Z}$ for $\mathbb{H}^2 \times \mathbf{R}$; a group with two components and identity component $(\mathbf{R} \times \widetilde{\mathrm{SL}}_2(\mathbf{R}))/\mathbf{Z}$ and point stabilizer $O(2,\mathbf{R})$ for $\widetilde{\mathrm{SL}}(2,\mathbf{R})$; the semidirect product of the $3$-dimensional Heisenberg group with $O(2,\mathbf{R})$ for $\mathrm{Nil}$; and the group of maps of $2$-dimensional Minkowski space to itself that are isometries or multiply the metric by $-1$ for $\mathrm{Sol}$.

Then: for every oriented prime closed $3$-manifold $M$ there is a finite collection $T \subset M$ of disjoint embedded incompressible $2$-tori such that each connected component of $M \setminus T$ has interior admitting a finite-volume geometric structure modelled on one of the eight geometries above.