Suppose $K$ is a number field and I have a subgroup of $GL_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group?
More precisely, the algorithm should tell me:
whether the group admits a finite presentation or not;
in case it does admit a finite presentation, it should exhibit one such presentation.
(For the purposes of this problem, let's assume $K$ is "computable", meaning that the computer knows a $\mathbb{Q}$-basis for it and the multiplications between those elements.)