Most manifolds are hyperbolic?

The quotes are from Thurston's survey paper Three dimensional manifolds, kleinian groups and hyperbolic geometry page 362:

2.6. THEOREM [Th 1]. Suppose $L \subset M^3$ is a link such that $M — L$ has a hyperbolic structure. Then most manifolds obtained from $M$ by Dehn surgery along $L$ have hyperbolic structures. In fact, if we exclude, for each component of $L$, a finite set of choices of identification maps (up to the appropriate equivalence relation as mentioned above), all the remaining Dehn surgeries yield hyperbolic manifolds.

 

Every closed 3-manifold is obtained from the three-sphere $S^3$ by Dehn surgery along some link whose complement is hyperbolic, so in some sense Theorem 2.6 says that most 3-manifolds are hyperbolic.

In two dimensions, most oriented manifolds are hyperbolic (since the only non-hyperbolic ones are $S^2$ and $T^2.$ In three dimensions, there are a number of models for random manifolds, and in most of them the vast majority of the manifolds obtained are hyperbolic. For example, a random mapping torus is hyperbolic, because a random surface automorphism is pseudo-Anosov (this is independently due to Joseph Maher and myself), a random Heegaard splitting is hyperbolic (Maher) (see my paper for more). It should be noted that people do not believe that these are "accurate" models of random 3-manifolds. As a negative statement, it is known that if you order knots by number of crossings, and you pick one of the first $N$ uniformly at random, then a random knot is not hyperbolic (there is a positive proportion of non-hyperbolic ones).

What is certainly true is that manifolds with hyperbolic-like properies are more common that one might naively suspect after taking a course in higher dimensional topology.

For example, the Gromov-Charney-Davis hyperbolization shows that for any closed smooth $n$-manifold $M$ there is a closed smooth $n$-manifold $N$ with (word)-hyperbolic fundamental group and a degree one map $f: N\to M$ such that $f$ is surjectve on homology and the fundamental group, and pulls back the rational Pontryagin classes. Ontaneda's recent work implies that for any $\epsilon>0$ the manifold $N$ can be chosen to admit a Riemannian metric of curvature within $[-1-\epsilon, -1]$.