The word problem for fundamental groups of smooth projective varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:30:28Z http://mathoverflow.net/feeds/question/22007 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22007/the-word-problem-for-fundamental-groups-of-smooth-projective-varieties The word problem for fundamental groups of smooth projective varieties Andy Putman 2010-04-21T03:05:49Z 2010-04-21T04:19:20Z <p>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 <a href="http://en.wikipedia.org/wiki/Word_problem_for_groups" rel="nofollow">word problem</a> for $\pi_1(X)$ solvable?</p> <p>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.</p> <p>EDIT : Another relevant remark is that the answers to the question <a href="http://mathoverflow.net/questions/15087/computing-fundamental-groups-and-singular-cohomology-of-projective-varieties" rel="nofollow">here</a> show that presentations for $\pi_1(X)$ are computable, so there are no issues there.</p> http://mathoverflow.net/questions/22007/the-word-problem-for-fundamental-groups-of-smooth-projective-varieties/22011#22011 Answer by Ben Wieland for The word problem for fundamental groups of smooth projective varieties Ben Wieland 2010-04-21T04:19:20Z 2010-04-21T04:19:20Z <p>Try <a href="http://www.ams.org/mathscinet-getitem?mr=1676613" rel="nofollow">Bogomolov and Katzarkov</a> (<a href="http://books.google.com/books?id=bJaRUqQEGMcC&amp;pg=PA85" rel="nofollow">google books</a> or <a href="http://www.ams.org/mathscinet-getitem?mr=1616131" rel="nofollow">an earlier paper</a>). 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.</p>