In [Gromov's talk at the Clay Math Research][1] from 23:23 to 25:55 Gromov says (slightly paraphrased)

> I want to emphasize a problem which
> comes from mathematical physics which
> is unsolved which is indicating that
> we don't understanding what
> triangulation is. And the problem is
> extremely simple. You take a manifold,
> and you just ask how many
> triangulations it has with a given
> number of simplices. So you have your
> manifold $X$ and you have the number
> of triangulations with k simplices
> $N_k(X)$ and you want to know what
> happens to it as $k$ goes to infinity,
> roughly. You take triangulations up to
> isomorphism. It's bounded below by $(1
> + \epsilon)^k$ and it's bounded from above by $k^k$, roughly. That's kind
> of trivial, you just keep subdividing
> and you see how many automorphisms you
> have an a $k$ element set. The
> question is, where is the truth?  And
> nothing is known, just absolutely
> blank. For surfaces you know, it's
> like that [exponential] and physicists
> kind of made that computation. 
> 
> [...] 
> 
> The whole point is that the manifold
> must be fixed, if you vary the
> manifold you would have $k^k$.
> 
> The subtle point is fix a topological
> manifold, how does the combinatorics
> tell you something about the topology.
> We think we understand it, but when we
> do this computation we don't. There's
> absolutely not a direct link between
> the two. We have a zero level question
> in topology, we cannot answer it.

The introduction to Kontsevich's thesis *Intersection theory on the moduli space of curves and the matrix Airy function* gives references to the solution to the problem for closed surfaces. Are there any other papers in the literature that discuss this problem?
 

  [1]: http://www.youtube.com/watch?v=u5DLpAqX4YA&feature=BFa&list=PL0767A09CF0864F8A&lf=plpp_video