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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
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@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

2 Answers 2

up vote 10 down vote accepted

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

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You beat me by 5 seconds! That'll teach me not to spend a minute being careful to spell people's names correctly... –  Andy Putman May 7 '10 at 14:40
    
:) I was worried that adding the sentence about recent progress would make me the loser. Wheh. –  Richard Kent May 7 '10 at 14:41

At least for the Weil-Petersson metric on Teichmuller space, this is a well-known open problem known as the Ehrenpreis conjecture. It has a rather fearsome reputation.

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I arbitrarily selected Richard's answer, and upvoted both. Should this conjecture be "metric-independent"? That is, if you know it for the WP metric, can you easily deduce it for the Teichmuller metric as well? –  David Hansen May 7 '10 at 14:48
    
Actually, the original problem is about the Teichmuller metric, and is harder than doing it for WP. –  Richard Kent May 7 '10 at 15:20
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Kahn and Markovic have solved the problem with respect to the "normalized WP metric" (which is the natural version of the WP metric to take). –  Sam Nead May 7 '10 at 22:00

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