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?



Is the diagonal of finitely presented groups computable?
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:
> 
> 
> 1. the kernel of $G\to H$ is finitely generated;
> 2. $G$ itself is finitely presented;
> 3. $H$ 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](https://arxiv.org/abs/1003.5117). 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.


