# Most manifolds are hyperbolic?

I heard the claim as in the title for a long time, but can not find the precise reference for this claim, what's the reference with proof for this claim? Thanks for the help. To be more precise, is there a canonical topology structure on the space $\Omega$ of all compact $n$-dim smooth manifolds, such that for any compact smooth $n$-dim manifold $M^n$, any neighborhood $U$ of $M^n$ in $\Omega$, there is a $n$-dim smooth manifold $N^n\in U$ such that $N^n$ admits a Riemannian metric with curvature equal to $-1$. Everything in my mind is just Riemannian hyperbolic, no complex structure involved.

• Sounds as meaningful as "most sets are groups". – YCor Dec 3 '16 at 1:47
• @YCor: The question is whether most manifolds (in some suitably defined sense) can be given a hyperbolic structure. That's very different than "most sets are groups." In 3 dimensions one could say that the fraction of closed $3$-manifolds built from $k$ $3$-simplices which are hyperbolic approaches $1$ as $k\to\infty$. – Jim Conant Dec 3 '16 at 2:25
• Again on Wikipedia: "On the other hand the heuristic statement that 'a generic 3–manifold tends to be hyperbolic' is verified in many contexts." en.wikipedia.org/wiki/Hyperbolic_3-manifold – Sam Hopkins Dec 3 '16 at 5:19
• Let's not vote to close it right away, but rather encourage the author to dig a little deeper. What does hyperbolic mean? I anticipate that an affirmative answer in the spirit of the assertion is constrained to low dimensions (maybe at most four). – Todd Trimble Dec 4 '16 at 4:15
• @JimConant: It is indeed conjectured that "the fraction of closed 3-manifolds built from $k$ 3-simplices which are hyperbolic approaches 1 as $k\to\infty$", however, to the best of my knowledge, this is still open. There are positive results in this direction though mentioned by Igor Rivin in his answer. – Moishe Kohan Dec 4 '16 at 13:20

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 $[-1-\epsilon, -1]$.

Some answers here explain why in small dimensions being hyperbolic is generic. Maybe surprisingly, in some sense being hyperbolic is actually rare in higher dimensions (while in other senses it is generic, as Igor Belegradek explains in his answer).

In "Counting hyperbolic manifolds" Burger-Gelnader-Lubotzky-Mozes proved that for $d\geq 4$ there is a constant $c(d)$ such that for $V$ large enough, the number of $d$-dimensional hyperbolic manifolds of volume at most $V$ is bounded by $V^{c(d)V}$.

Note that by Mostow Rigidity, a given manifold carries at most one hyperbolic structure, and if it does, this manifold is completely determinded by its fundamental group. What BGLM do is actually counting possible fundamental groups.

On the contrary, in dimension 3 there exist infinitely many different (pairwise non-homeomorphic) compact hyperbolic manifolds of uniformly bounded volume (examples could be obtained by Dehn filling a knot complement). By taking a product with a $(d-3)$-dimensional torus, one sees that also changing "hyperbolic" to "non-positively curved" (sectional curvature in $[-1,0]$) gives infinitely many compact manifolds of uniformly bounded volume.

Maybe I should (and maybe I shouldn't) note that one can generalize the above counting result to the realm of all pinched negatively curved manifolds of dimension $\geq 5$. We do this in a forthcoming paper with Gelander and Sauer. For this one should relay on Farrel-Jones Theorem instead of Mostow Rigidity.

• Is there some way to compare to these growth rates of hyperbolic manifolds to growth rates of the number of non-hyperbolic manifolds? – Tim Campion Dec 5 '16 at 9:12
• @TimCampion I added a paragraph regarding dim=3. Note that by taking product with $S^1$ (say) you get infinitely many 4-manifolds of uniformly bounded volume. – Uri Bader Dec 5 '16 at 9:26