# Covers of Riemann surfaces which become arbitrary close in Teichmuller space

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.

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I'm wondering whether there is any motivation for this question other than just curiosity? – Kevin H. Lin May 7 '10 at 14:31
@Kevin: Rick drew an analogy to certain questions in graph theory (!), which unfortunately I don't remember well enough to repeat here. – David Hansen May 7 '10 at 14:39
@David : I guess the kind of graph theory analogous result is that any two finite connected $d$-regular graphs (for some $d>2$) have a common finite covering (Angluin's theorem). Actually, there is a simple group-theoretic proof, using that a finite connected $d$-regular graph is the same thing as a conjugacy class of finite index torsion-free subgroup of the free product $\mathbb{Z}/2\star \mathbb{Z}/d$. – BS. Jul 16 '11 at 12:12
ADDED : there is a more general result that any two finite graphs having the same universal cover have a finite common cover (Leighton's theorem). This extends to colored graphs, see math.columbia.edu/~neumann/preprints/leighton1.5.pdf – BS. Jul 16 '11 at 12:36

This is the Ehrenpreis Conjecture, and is still open.