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While attending a very nice talk on the geometric group theory of fundamental groups of Kahler manifolds by Pierre Py last weekend, I realized that I don't know the answer to the following question. Let $X$ be a smooth projective variety over $\mathbb{C}$. Is the word problem for $\pi_1(X)$ solvable?

Here are a couple of relevant facts. Taubes proved that every finitely presentable group is the fundamental group of a compact complex manifold of complex dimension 3. Earlier, Gompf proved that every finitely presentable group is the fundamental group of a compact symplectic manifold of real dimension 4. Thus the word problem is not solvable for fundamental groups of compact complex manifolds. Also, Toledo has an example of a smooth compact projective variety whose fundamental group is not residually finite. This rules out using maps to finite groups to solve the word problem, and also shows that $\pi_1(X)$ need not be linear.

EDIT : Another relevant remark is that the answers to the question here show that presentations for $\pi_1(X)$ are computable, so there are no issues there.

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Interestingly, I heard the same question from Dick Hain in a conversation about 15 years ago. But I have no feeling for whether it ought to be true or false. – Donu Arapura Apr 21 '10 at 11:38
up vote 4 down vote accepted

I was going through my old questions and realized that this one did not have a good answer (as far as I know, the reference that Ben Wieland gave in his answer does not work). I've since learned that it is a well-known open question. However, I thought I'd point out the recent paper

Kapovich, Michael, Dirichlet fundamental domains and topology of projective varieties. Invent. Math. 194 (2013), no. 3, 631–672.

which proves that every finitely presentable group is the fundamental group of an irreducible projective variety with very mild singularities (only normal crossings and Whitney umbrellas), and thus the word problem is unsolvable for the fundamental groups of such varieties. If you don't require irreducibility, the paper

Kapovich, Michael and Kollár, János, Fundamental groups of links of isolated singularities. J. Amer. Math. Soc. 27 (2014), no. 4, 929–952.

manages to do this with only normal crossing singularities.

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Try Bogomolov and Katzarkov (google books or an earlier paper). I don't understand the statements, but I think that for every finitely presented group, they find a extensions of surface groups by the given group that are "approximated" by projective groups. The quality of the approximations is not clear, but I suspect that they preserve uncomputability.

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Ben, could you (or any other passing expert!) explain the relationship between the Bogomolov--Katzarkov paper and the work of Delzant--Gromov on cuts in Kahler groups? – HJRW Apr 21 '10 at 17:11

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