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What are the properties that hold for a fundamental group of a surface and does not hold necessary for the fundamental groups of manifolds of higer dimensions ??

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"fundamental groups of manifolds of higer dimensions": every finitely presentable group is the fundamental group of some manifold of dim $\ge 4$. – André Henriques Jun 18 '11 at 10:39
In some sense, André's comment is the answer (see also Jim's answer below): closed 4-dimensional manifolds are "as bad" as finitely presented groups: so their fundamental group can be non-linear, non-residually finite, non-hopfian,etc...; it can have Kazhdan's property (T), it can have all $L^2$-Betti numbers vanishing, etc... I share Henri's opinion that the OP is vague. – Alain Valette Jun 19 '11 at 5:17

This question is very vague, but here are some thoughts to add to Mark's answer.

First, note that any finitely presented group arises as the fundamental group of a closed manifold of dimension 4 (see this MO question), which is a huge contrast to the very special case of dimension 2.

The properties of the fundamental groups of 3-manifolds are a subject of very active research, much aided by Perelman's solution to the Geometrisation Conjecture. Like the 2-dimensional case, 3-manifold groups are residually finite (a theorem of Hempel). The fact that there is no closed 3-manifold with every infinite-index subgroup free is only very recent known, as a result of work of Kahn and Markovic.

I don't think any closed 3-manifold has cohomological dimension 2, so that property actually does it on its own.

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It's a conjecture that surface groups are characterized by being the only 1-relator groups such that every finite-index subgroup is also 1-relator and every infinite index subgroup is free.

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Neat. Whose conjecture is that? – Ryan Budney Jun 19 '11 at 2:40
oops, I forgot a condition. Ben Fine: – Ian Agol Jun 19 '11 at 4:33
FYI, I recently proved this for 'cyclically pinched' one-relator groups: . – HJRW Jun 19 '11 at 6:33
Hey Henry, I haven't thought about this much - why is the condition on infinite index subgroups being free necessary? Is there a (residually finite) 1-relator group with finite-index subgroups 1-relator, but non-free infinite-index subgroups? – Ian Agol Jun 19 '11 at 23:34
Ian - The Baumslag--Solitar group BS(1,m) has this property for any m. In fact, this question goes back to Melnikov in the Kourovka notebook (though BS(1,m) is a counterexample to Melnikov's original question). – HJRW Jun 20 '11 at 6:01

Every subgroup of infinite index is free, the group is residually finite, and the cohomological dimension is 2.

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how do you prove thay every subgroup of infinite index is free ? – unkown Jun 18 '11 at 9:57
It is well known: A. Hoare, A. Karrass and D. Solitar, Subgroups of infinite index in Fuchsiangroups, Math. Z. , 125, 1972, 59–69 – Mark Sapir Jun 18 '11 at 10:11
unknown - consider the corresponding covering space, and convince yourself that it deformation retracts onto a graph. – HJRW Jun 18 '11 at 10:38
@HW: your statement is actually a nontrivial theorem of Whitehead, so you might be overestimating the OP's mathematical prowess. – Igor Rivin Jun 19 '11 at 0:39
Igor - I suppose I really had the finitely generated case in mind. – HJRW Jun 19 '11 at 6:35

The word problem for the fundamental group of a closed surface is solvable, using Dehn's algorithm. Since any finitely presented group appears as the fundamental group of some closed $4$-manifold, and there are such groups for which the word problem is unsolvable, this is indeed a special property for two dimensions.

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Not really special to two dimensions, since also true in three dimensions. – Igor Rivin Jun 19 '11 at 2:18
@Igor: I didn't interpret the question that way. I interpreted it to mean, what's true in two dimensions that's not true in general. – Jim Conant Jun 19 '11 at 12:22
@Jim: I personally think the question is rather poor, so any answer makes more sense than the question itself. To increase the silliness slightly: "what is true of free groups which is not true of general groups"? – Igor Rivin Jun 20 '11 at 0:49

A surface group is either virtually abelian, or word hyperbolic (or both, when it is finite).

In some sense, this reflects the fact that every surface admits a Riemannian metric of constant curvature, and that the sign of the curvature is detected by the fundamental group.

In dimension 3, Perelman's uniformization implies that compact manifolds can be decomposed into "geometric pieces" (that are again detected in a suitable sense by their fundamental groups), while in higher dimension there is no hope for a simple result of this type.

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