# Judging whether a finitely presented group is a 3-manifold group?

Given a finitely presented group $G$, how many necessary conditions do people know for $G$ to be isomorphic to the fundamental group of some closed connected 3-manifold? (e.g. residually finite)

• How do you tell whether a finitely presented group is residually finite? It is hard enough to tell if it is trivial... – Igor Rivin Oct 22 '12 at 2:49
• @Li: There are way too many restrictions: Coherence, virtual cohomological dimension $\le 3$, every infinite amenable subgroup is polyciclic; either virtually $PD(3)$ or (virtually) splits as a free product; either contains $Z^2$ or is hyperbolic; either finite or has virtually infinite abelianization, etc. You may want to narrow down the scope of the algebraic properties you are interested in. As Igor said, most of these properties are hard (or impossible) to verify algorithmically. – Misha Oct 22 '12 at 3:00
• I know it is difficult to use them in practice. But I just want to know some such kind of conditions. – Li Yu Oct 22 '12 at 3:10
• The word problem is solved for 3-manifold groups, so whatever conditions you'd have to check, they'd be non-computable in practice. – Ryan Budney Oct 22 '12 at 3:13
• It seems to me that this question is asking for a list, and hence should be community wiki. – HJRW Oct 22 '12 at 9:38

Apologies for the shameless self-promotion, but as you ask for necessary conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.

Aschenbrenner, Friedl and I have given what I think is a pretty exhaustive list in our survey paper here.

• Yes, that is what I'd like to know. – Li Yu Oct 22 '12 at 10:09
• Why not this paper? arxiv.org/abs/1210.2101v1 – Ian Agol Oct 22 '12 at 11:50
• Well, I thought the question was asking for an exhaustive list, so the survey paper seemed like the 'right' answer. But if you actually want to recognise whether or not something's a 3-manifold group then, yes, that paper is the one to look at! – HJRW Oct 22 '12 at 13:21
• Regarding the main theorem of GMW, that recognizing presentations of 3-manifold groups is reducible to the word problem; doesn't this follow from general theorems about c.e. sets? The word problem is equivalent to the membership problem for a universal c.e. set. Deciding whether a presentation is of a 3-manifold group is equivalent to the membership problem for a particular c.e. set, because one can list all 3-manifolds (say, as simplicial complexes) and hence the presentations of their groups. Hence the latter problem is reducible to the word problem, or am I missing something? – John Stillwell Oct 25 '12 at 22:47
• @John Stillwell: See Remark 2.6 of our paper, in which we make (I believe) exactly the point that you do in your comment(s) above. We mean something different by `recursive modulo the word problem'. See Definition 2.4 (and proceeding definitions) for a precise statement of what we mean. – Daniel Groves Oct 26 '12 at 2:30

since Henry started the shameless self-promotion, let me also do so... Given any group $\pi$ one can study the corresponding Alexander polynomial $\Delta_\pi$ which lies in the group ring of $H:=H_1(\pi;\Bbb{Z})/\mbox{torsion}$. If $\pi$ is the fundamental group of a closed 3-manifold, then the Alexander polynomial $\Delta_{\pi}$ is symmetric and the one-variable specializations have even degree. (see F, Kim, Kitayama: Poincaré duality and degrees of twisted Alexander polynomials)

The symmetry holds also if $\pi$ is a 3-dimensional Poincare duality group, but I am not sure whether the degree condition holds in that case.

The advantage is that this condition can be checked easily, and by checking it for finite index subgroups one gets even more necessary conditions. I would guess that in practice this is a very effective way for weeding out non 3-manifold groups.

At least it allowed me to make the right bet on Ryan's example...

As a demonstration of difficulty, a counter-question: Is this a 3-manifold fundamental group, and if so, which one?

$$\langle a, b | a^2b^{-1}a^{-2}ba^{-1}ba^{-2}b^{-1}a^2b, a^{-1}b^{-1}a^2ba^{-4}ba^2b^{-1}a^{-1}b^{-1}a^2ba^{-2}ba^2b^{-1} \rangle$$

• The program Heegaard (written by Berge) says the presentation is realizable. So yes, it is a three-manifold group. Also, the homology is Z. If pressed I would take the diagram that his program produces, convert that to a triangulation, and feed the result to SnapPy. – Sam Nead Oct 26 '12 at 4:54
• If I haven't made a mistake: I used a variety of tools, ending with snappy, to obtain a triangulation of this three-manifold M. I then used regina to find fundamental normal tori. Cutting proves that M is obtained by gluing X, the figure eight knot complement, to Y, the Seifert fibered space with base orbifold D(2,2) (aka the orientation I-bundle over the Klein bottle). Unfortunately, regina doesn't track gluing data when we cut along a normal torus. So we can't recover the gluing data that way. – Sam Nead Jan 10 '16 at 10:15
• Thinking about homology considerations tells us that the longitude of X is glued to the longitude of Y - that reduces the set of possible gluings to a "line" of Dehn twists. I think that your "challenge" could now be answered computationally if we had a facility to glue manifolds with torus boundary. Does that exist? – Sam Nead Jan 10 '16 at 10:49