OK, I will also give it a shot.

First of all, I don't like to sell Geometrization because it helps with the homeomorphism problem. The Geometrization Theorem is an object of stunning beauty (most' "most 3-manifolds are hyperbolic" should be a convincing an exciting statement for anybody in an elevator who has sign seen the art of M.C.Escher), and beauty in mathematics is usually a sign that we are on the right track. And indeed, the beauty of Geometrization begets all kinds of results.

Regarding the new results of Agol, Wise et al. one should perhaps not jump right to "virtual Haken" or "virtually fibered" but one should look at the "real theorem", the Virtually Compact Special Theorem which goes as follows:

If $N$ is a finite volume hyperbolic 3-manifold, then $\pi_1(N)$ is virtually compact special, i.e. $\pi_1(N)$ is virtually a quasi-convex subgroup of a Right Angled Artin Group (RAAG).

One can explain a RAAG to anybody who has seen group theory in an elevator between about 3 floors. The fact that "simple" objects like RAAGs contain all hyperbolic 3-manifold groups (up to going to a finite index subgroup) is stunning and beautiful. All the goodies, e.g. largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc come from that statement (well, together with Agol's fibering theorem, tameness etc.)etc.). This can be seen clearly by looking at Diagram 4 in a recent survey paper on 3-manifold groups by authors whose names escape me at the moment. It is really stunning how the Virtually Compact Special Theorem answers all open questions at once. It is one of the great achievements of Dani Wise to have found the "right statement".

(Note that largeness, linear over $\mathbb{Z}$, biorderable do NOT follow from virtual fibering or virtual Haken alone.)

Back to the elevator:

The results make me think that hyperbolic 3-manifolds are like Jack in the Box. If you take a hyperbolic integral homology sphere you look at a tiny manifold, but when you press a button (i.e. go to an appropriate finite cover), the 3-manifold suddenly becomes a grand object of beauty (e.g. has as many fibered faces in the Thurston norm ball as you could wish).

(This analogy also works with tiny seed, a bit of water, blooming flower etc. for the botanically minded elevator companion)

So to conclude, I think the Geometrization Theorem and the Virtually Compact Special Theorem of Agol-Wise are stunningly beautiful results. The fact that the statements are so beautiful made it highly plausible that they were right, even before they were proved (I can't imagine that any serious person doubted the Poincare conjecture after Thurston stated the Geometrization conjecture). And ideally it's this beauty which I would like to communicate.

1

OK, I will also give it a shot.

First of all, I don't like to sell Geometrization because it helps with the homeomorphism problem. The Geometrization Theorem is an object of stunning beauty (most' 3-manifolds are hyperbolic should be a convincing statement for anybody in an elevator who has sign the art of M.C.Escher), and beauty in mathematics is usually a sign that we are on the right track. And indeed, the beauty of Geometrization begets all kinds of results.

Regarding the new results of Agol, Wise et al. one should perhaps not jump right to "virtual Haken" or "virtually fibered" but one should look at the "real theorem": If $N$ is a finite volume hyperbolic 3-manifold, then $\pi_1(N)$ is virtually compact special, i.e. $\pi_1(N)$ is virtually a quasi-convex subgroup of a Right Angled Artin Group (RAAG). One can explain a RAAG to anybody who has seen group theory in an elevator between about 3 floors. The fact that "simple" objects like RAAGs contain all hyperbolic 3-manifold groups (up to going to a finite index subgroup) is stunning and beautiful. All the goodies, e.g. largeness, linear over $\mathbb{Z}$, virtual fibering, LERF, virtually biorderable etc come from that statement (well, together with Agol's fibering theorem, tameness etc.)

(Note that largeness, linear over $\mathbb{Z}$, biorderable do NOT follow from virtual fibering or virtual Haken alone.)

The results make me think that hyperbolic 3-manifolds are like Jack in the Box. If you take a hyperbolic integral homology sphere you look at a tiny manifold, but when you press a button (i.e. go to an appropriate finite cover), the 3-manifold suddenly becomes a grand object of beauty (e.g. has as many fibered faces in the Thurston norm ball as you could wish).

(This analogy also works with tiny seed, a bit of water, etc. for the botanically minded elevator companion)

So to conclude, I think the Geometrization Theorem and the Virtually Compact Special Theorem of Agol-Wise are stunningly beautiful results. The fact that the statements are so beautiful made it highly plausible that they were right, even before they were proved (I can't imagine that any serious person doubted the Poincare conjecture after Thurston stated the Geometrization conjecture).