I am wondering about the following:



> 
> Suppose that $S$ is a non-compact
>  hyperbolic surface of finite area. 
>  Suppose that $\lambda \subset S$ is a
>  non-trivial, geodesic,
>  measured lamination. Forget the transverse measure. Is there a
>  (non-compact) geodesic lamination $\lambda'$
>  containing $\lambda$, and a sequence of
>  ideal geodesic triangulations $T\_i$, so
>  that $T\_i \to \lambda'$ in the
>  Chabauty topology?
> 
> 
> 


A little bit of context:


* The Chabauty topology is a generalization of the Hausdorff topology to non-compact sets. It is characterized by the condition that every point of $\lambda'$ is the limit of a sequence of points $x\_i \in T\_i$, and conversely every convergent sequence $x\_{i\_n} \in T\_{i\_n}$ limits to a point of $\lambda'$. See, e.g. [Notes on Notes of Thurston](http://books.google.com/books?id=MoZlcJv6ww0C&lpg=PR5&dq=canary%20epstein%20green&pg=PA60#v=onepage&q=chabauty&f=false).
* I am mainly interested in the answer in the setting where $\lambda$ is the pleating lamination on the boundary of the convex core of a quasifuchsian $3$-manifold. This places some additional hypotheses on $\lambda$: for example, it would have to be compact. But the question seems to be intrinsically $2$--dimensional, and it's not clear to me how to use compactness of $\lambda$ as a hypothesis.
* If the ideal triangulations $T\_i$ are replaced by simple closed curves, the result is well-known. So one approach would be to take a sequence of closed curves $C\_i$, limiting to $\lambda' \supset \lambda$, approach each $C\_i$ by triangulations (twisting more and more), and then take a diagonal sequence of triangulations. But it's not clear that this diagonal sequence even converges.


Anyway, either a reference or a way to argue would be much appreciated!



Laminations as a limit of ideal triangulations
It turns out that the answer to the question is "yes". Saul Schleimer and I needed this result for a paper that we just finished writing, so we ended up sorting it out. The full argument is written down in [Lemma A.6 in the Appendix of this paper](https://arxiv.org/abs/1108.5748), so what follows below is an outline.


Take a sequence of closed curves $C\_i$, which limits to $\lambda$ in the measure topology. This sequence has a subsequence (which I will still call $C\_i$) limiting to $\lambda' \supset \lambda$ in the Chabauty topology. Now, each $C\_i$ is contained in the Chabauty limit of a sequence of triangulations $T\_{i,j}$. This means that one can take a representative triangulation $T\_{i,j(i)}$ that is very close to $C\_i$, where "very close" can be quantified (say, closer than distance $1/i$) because the Chabauty topology is metrizable. Now, the sequence $T\_{i,j(i)}$ will converge to a lamination $\lambda'' \supset \lambda' \supset \lambda$.


