Let $f:G\to H$ be a surjective homomorphism of finitely presented groups. If the kernel of $f$ is finitely generated then is $G\times_H G$ is a finitely presented group? Can one compute an explicit finite presentation?
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$\begingroup$ If $F$ is free of finite rank and $Q$ is an infinite quotient of $F$ with infinite kernel, then the fiber product $F\times_Q F$ is not finitely presentable. $\endgroup$– YCorCommented Apr 22, 2021 at 8:00
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$\begingroup$ @YCor I edited is it correct now? $\endgroup$– user178109Commented Apr 22, 2021 at 8:04
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1$\begingroup$ If you checked that the fiber product is finitely presentable, don't you get an explicit finite presentation? $\endgroup$– YCorCommented Apr 22, 2021 at 8:13
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2$\begingroup$ @YCor: the fact that they are finitely presented follows from the 1-2-3 theorem of Baumslag--Bridson--Miller--Short, but the proof doesn't give an explicit presentation. $\endgroup$– HJRWCommented Apr 22, 2021 at 8:48
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1$\begingroup$ @YCor: I omitted to mention that there is an extra hypthesis needed, namely that the quotient $H$ should be of type $F_3$. $\endgroup$– HJRWCommented Apr 22, 2021 at 20:35
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1 Answer
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The answer is "no". The hypotheses for the fibre product to be finitely presentable are given by the 1-2-3 theorem of Bridson--Baumslag--Miller--Short:
1-2-3 Theorem (BBMS): The fibre product $G\times_H G$ is guaranteed to be finitely presentable as long as:
- the kernel of $G\to H$ is finitely generated;
- $G$ itself is finitely presented;
- $H$ satisfies a higher finiteness condition, namely being of "type $F_3$".
However, there is no general algorithm that can compute a finite presentation for $G\times_H G$. Indeed, stronger, there is no algorithm that can compute its abelianisation, and the result follows since the abelianisation can be computed from a finite presentation.
This is proved in my paper with Bridson:
Bridson & Wilton, "On the difficulty of presenting finitely presentable groups", Groups Geom. Dyn. 5(2), pp. 301–325, 2011
available on the arXiv here. See Theorem A, and note in the proof that the subgroup $\Lambda_n$ is indeed a fibre product.
To give a brief sketch of the argument, one takes a sequence of perfect groups $Q_n$ such that the second Betti numbers $b_2(Q_n)$ are known to be impossible to compute. The Rips short exact sequence gives
$$1\to K_n\to\Gamma_n\to Q_n\to 1$$
with $K_n$ finitely generated, and then let $\Lambda_n\leq \Gamma_n\times\Gamma_n$ be the fibre product. Finally, an argument with the Stallings exact sequence relates the Betti numbers:
$$b_1(\Lambda_n) = 2b_1(\Gamma_n)+b_2(Q_n).$$
Since $b_1(\Gamma_n)$ is computabe and $b_2(Q_n)$ isn't, it follows that $b_1(\Lambda_n)$ is not computable. In particular, no presentation for $\Lambda_n$ can be computed, even though we are given explicit generating sets and know that they are finitely presented.
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$\begingroup$ So then at which level in the arithmetical hiearchy is the claim that a given group is finitely presentable? Seems like an oracle for the halting problem couldn't solve that? $\endgroup$– user178109Commented Apr 22, 2021 at 11:00
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2$\begingroup$ @Oniqa: that question is above my pay-grade, unfortunately! $\endgroup$– HJRWCommented Apr 22, 2021 at 11:51
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1$\begingroup$ @Oniqa What data are you using to represent the group? Are we talking about arbitrary computable groups? Or do you mean relative to the group as an oracle? $\endgroup$ Commented Apr 23, 2021 at 4:40
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$\begingroup$ @Oniqa I guess it doesn't really matter that much. Suppose you're given a group as an oracle $G$. If you pick some finite set $\bar{a}$ of elements and some relations on those, checking to see if those relations give a presentation of the subgroup generated by $\bar{a}$ is a $\Pi^0_1(G)$ condition. $\endgroup$ Commented Apr 23, 2021 at 5:03
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3$\begingroup$ @Oniqa Checking to see if a finite presentation exists is therefore $\Sigma^0_3(G)$, which implies that checking to see if no such finite presentation exists is $\Pi^0_3(G)$. So all told, deciding if a given group has a finite presentation takes at most $3$ Turing jumps above the complexity of the group itself. I don't know if this is optimal, but usually these kinds of things aren't any easier than a naïve calculation like this. (Although I also may have miscounted.) $\endgroup$ Commented Apr 23, 2021 at 5:04