Judging whether a finitely presented group is a 3-manifold group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:07:34Z http://mathoverflow.net/feeds/question/110293 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110293/judging-whether-a-finitely-presented-group-is-a-3-manifold-group Judging whether a finitely presented group is a 3-manifold group? Li Yu 2012-10-22T02:38:39Z 2012-10-25T22:08:13Z <p>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)</p> http://mathoverflow.net/questions/110293/judging-whether-a-finitely-presented-group-is-a-3-manifold-group/110300#110300 Answer by Ryan Budney for Judging whether a finitely presented group is a 3-manifold group? Ryan Budney 2012-10-22T04:15:47Z 2012-10-22T04:15:47Z <p>As a demonstration of difficulty, a counter-question: Is this a 3-manifold fundamental group, and if so, which one? </p> <p>$$\langle a, b | a^2b^{-1}a^{-2}ba^{-1}ba^{-2}b^{-1}a^2b, a^{-1}b^{-1}a^2ba^{-4}ba^2b^{-1}a^{-1}b^{-1}a^2ba^{-2}ba^2b^{-1} \rangle$$</p> http://mathoverflow.net/questions/110293/judging-whether-a-finitely-presented-group-is-a-3-manifold-group/110311#110311 Answer by HW for Judging whether a finitely presented group is a 3-manifold group? HW 2012-10-22T09:38:47Z 2012-10-22T10:42:57Z <p>Apologies for the shameless self-promotion, but as you ask for <em>necessary</em> conditions, you seem to want a list of theorems of the form 'If G is a 3-manifold group then G has property P'.</p> <p>Aschenbrenner, Friedl and I have given what I think is a pretty exhaustive list in our survey paper <a href="http://arxiv.org/abs/1205.0202" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/110293/judging-whether-a-finitely-presented-group-is-a-3-manifold-group/110707#110707 Answer by Stefan Friedl for Judging whether a finitely presented group is a 3-manifold group? Stefan Friedl 2012-10-25T22:08:13Z 2012-10-25T22:08:13Z <p>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)</p> <p>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.</p> <p>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.</p> <p>At least it allowed me to make the right bet on Ryan's example...</p>