Naive Reidemeister-Schreier for $\mathbb Z$ quotients

I have a question about a "standard" variant of the Reidemeister-Schreier algorithm used by topologists when manipulating manifolds they either know or suspect are fibre-bundles over $S^1$.

Say you have a manifold $M$, and you have computed a finite presentation for $\pi_1 M$, and you also have an onto homomorphism $\pi_1 M \to \mathbb Z$. We want to determine if the kernel is a finitely-presented group, and if it is, to write $\pi_1 M$ explicitly as a semi-direct product of this group with $\mathbb Z$.

One way to attempt this would be to lift a CW-structure for $M$ up to the corresponding covering space, then collapse a maximal tree in the $1$-skeleton. This gives an infinite presentation of the fundamental group of the homotopy fibre of the corresponding map $M \to S^1$. At this point (in practice) people just follow their noses and observe that this infinite presentation reduces to a finite sub-presentation. For example, you could modify your presentation of $\pi_1 M$ so that precisely one of your generators is mapped to a non-trivial element of $\mathbb Z$. Your maximal tree is then just concatenations of the lift of this non-trivial element's 1-cell. Then the game becomes looking at the lift of the 2-cells, and hoping you can find such 2-cells that allow you to do 1-cell cancelling moves. To do this quite literally you need 2-cells whose lift contain unique instances of "highest weight" 1-cells (similarly unique lowest weight). This would allow you to mimic (quite literally) things like Buchburger's algorithm for finding Groebner basis, to inductively remove all but finitely-many 1-cells and get to a finite presentation. Perhaps some of your 1-cells can't be eliminated because your relators do not have them as unique highest/lowest weight instances, but then you could perhaps modify your relators to find new ones that do. So one could make this into a fairly elaborate game.

Are there known more sophisticated algorithms out there, or do you know if people tend to find the above sufficient for most circumstances? Perhaps people know the above is essentially the best one can do? Any references would be helpful.

• I'm not sure if you care, but it follows from Bestvina--Brady's work on combinatorial Morse theory that determining if a cyclic cover of a non-positively curved cube complex is finitely presentable is undecidable. Perhaps it's possible to upgrade this to the manifold setting using Davis-style Coxeter-group techniques. – HJRW Sep 14 '13 at 6:38
• I believe its uncomputable. And I want to use this algorithm on 4-manifold fundamental groups... So I can't avoid those problems. What I want to know is "can I get away with any more than I think I can?" – Ryan Budney Sep 14 '13 at 16:28
• Right! I should have said 'Perhaps it's possible to upgrade this to the aspherical manifold setting using Davis-style Coxeter-group techniques.' – HJRW Sep 15 '13 at 7:33
• This is a comment on other methods outside of Reidemeister-Schreier. There is a dynamical systems method, based on "asymptotic cycles" or "homology directions" of flows, under which one shows that if you have one homomorphism to $\mathbb Z$ which fibers then so do all projectively nearby ones. This was used by Fried in his article in "Travaux de Thurston sur les surfaces", for understanding the dynamics of fibered facts of the Thurston norm on $H_2$ of a 3-manifold, but it applies more generally. – Lee Mosher Sep 15 '13 at 13:36