First of all, I am sorry if my question is obvious or it has already been explained somewhere. My question is
Apart from Poincare conjecture what are the other (important) applications of Thurston's geometrization conjecture?
2026-03-27 16:25:14.1774628714
What are the (important) applications of Thurston's geometrization conjecture?
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A closed orientable 3-manifold with finite fundamental group is diffeomorphic to a quotient of the 3-sphere by a group of spherical isometries acting freely. As far as I know, this was a well known conjecture before Perelmans proof of geometrisation.
To see roughly how it follows note that finite fundamental group rules out a non trivial JSJ decomposition with geometric components. Now the result follows by noting that the only Thurston geometry with a compact model space is the three sphere with the round metric.
Another application that comes to mind is that there exists an algorithm to test whether the fundamental group of two closed orientable 3-manifolds are isomorphic.