Suppose $S$ and $S'$ are two compact Riemann surfaces of genus $g$. Does there exist a sequence of genera $g\_i \to \infty$ and covers $S\_i, S\_{i}'$ of $S,S'$, both of genus $g\_i$, such that $d(S\_i,S\_{i}')\to 0$? Here $d$ a "natural" distance function on Teichmuller space, of which I suppose there are many, but for definiteness let's take it to be induced by the Teichmuller metric.


This question was asked to me by Rick Kenyon last year, and some brief thought on it got me nowhere.



Covers of Riemann surfaces which become arbitrary close in Teichmuller space
This is the Ehrenpreis Conjecture, and is still open.


Jeremy Kahn and Vlad Markovic have made some progress recently.


UPDATE: Kahn and Markovic have now announced a proof of the entire conjecture. See <http://arxiv.org/abs/1101.1330>


