Let $G$ be a group given by a finite presentation.
On the one hand, it is easy to determine the abelian invariants of $G$, or in other words,
it is algorithmically decidable whether $G$ surjects to a cyclic group of prime order.
On the other, it is known to be algorithmically undecidable whether $G$ has a finite
quotient at all. Therefore there is at least one series of finite simple groups
for which it cannot be decided by means of algorithms whether it contains a homomorphic
image of $G$ or not.



> 
> **Question:** For which series of finite simple groups is it algorithmically
>  decidable whether they contain a homomorphic image of a given finitely presented group? 
> 
> 
> 



For which series of finite simple groups is it algorithmically decidable whether they contain a homomorphic image of a given finitely presented group?
**UPDATE 22/10/17:** The question is answered for many classes of simple quotients in [this](https://arxiv.org/abs/1710.07183) preprint of Bridson--Evans--Liebeck--Segal.




---


Here's what I know.


Martin Bridson and I proved the [theorem](http://arxiv.org/abs/1401.2273) mentioned in the question, namely that it is undecidable whether or not a fp group has a non-trivial finite quotient. As you say, it follow trivially that, for at least one infinite family $S$ of finite simple groups, it's undecidable whether or not a fp group has quotient from $S$, and so the question is raised of which families $S$ this is true for.


I have seen a very early preprint which resolves this in several cases. Since it isn't publicly available, I don't want to reveal the authors, but, iirc, it proves that this is undecidable for the alternating groups, and also for any family of classical Lie groups (eg PSL(n,q) for any q as n varies).


I believe that techniques of proof are expected to extend to the other families of groups of Lie type, and this may be work in progress.


Remarkably, although I don't recall the reference, it is known that the problem is *decidable* when $n=2$ and $q$ varies (that is, when $S=\{PSL(2,q)\}$), and I believe this also for the other classical groups. Based on this, Bridson has conjectured (in conversation) that the same holds for any fixed $n$ when $q$ varies (and similarly for the other classical groups).


