Hempel proved that Haken manifolds have residually finite fundamental groups. With the Geometrization conjecture, this now holds for any compact and orientable 3-manifold.
How exactly does the Geometrization conjecture imply that the only non-Haken compact orientable irreducible 3-manifolds are compact hyperbolic manifolds with no cusps?
Thanks a lot.
There are a few ways to look at this, but let me give a synopsis of Peter Scott's discussion in Section 6 of his 8 geometries paper, because it seems to directly address the question:
Scott, Peter, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15, 401-487 (1983). ZBL0561.57001.
(Note: there is also an errata for this reference: http://www.math.lsa.umich.edu/~pscott/errata8geoms.pdf. The corrections mentioned there do not affect this summary. )
Let's set up some standard 3-manifold terminology (anything left out is clearly defined in the reference).
A compact orientable 3-manifold is irreducible if is either $S^1 \times S^2$ or every embedded $S^2$ bounds a 3-ball. From now on, assume $M$ is a compact, orientable, irreducible 3-manifold (unless otherwise specified).
A surface $S$ (smoothly) embedded in $M$ is incompressible if every embedded curve $\gamma$ in $S$ which bounds a disk in $M$ also bounds a disk in $S$. A compact, orientable, irreducible 3-manifold is toroidal if it contains an incompressible torus.
The most natural thing to say is that the affirmative solution to the Geometrization Conjecture implies that a compact, irreducible 3-manifold is either toroidal or is homeomorphic to a quotient of the form $X/\Gamma$ where $X$ is one of the following geometric spaces: $S^3,S^1 \times \mathbb{R},E^3,Nil,Sol, H^2 \times \mathbb{R}, \widetilde{PSL(2,\mathbb{R})}$ or $H^3$ and $\Gamma \subset Isom^+(X)$ acting properly and discontinuously.
If a space admits a Sol geometry it is Haken. In fact, it is both toroidal and has positive first Betti number, (see [Scott, Theorem 4.17]).
The remaining geometries are either $H^3$ or Seifert fibered. Of course, some Seifert fibered manifolds like $T^3 \cong S^1 \times S^1 \times S^1$ are both toroidal and Seifert fibered and there are other minor pathologies: for example, $RP^3 \# RP^3$ is Seifert fibered and reducible and in the wake of Geometrization manifolds with $S^3$ geometry are exactly those with finite fundamental group.
Happily, Scott gives a clean statement of what you want in the conjecture on page 484 (of course the affirmative solution to the Geometrization Conjecture implies this conjecture is now known to be true):
Conjecture (now theorem): Let $M$ be a closed, irreducible, non-Haken 3-manifold with infinite fundamental group. Then $M$ is either a Seifert fibered space or admits a hyperbolic structure.
To connect this statement to the comments mentioned above, if we assume $M$ is Seifert fibered, non-Haken, irreducible, and has infinite fundamental group, then it is small Seifert fibered, which implies $M$ that the base orbifold of $M$ is $S^2(a_1,a_2,a_3)$ with $1/a_1 + 1/a_2 + 1/a_3 \leq 1$.
