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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:

1. whether the group admits a finite presentation or not;

2. 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.)

Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ 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:

1. whether the group admits a finite presentation or not;

2. 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.)

Suppose 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?

(In this question, $K$ is a number field.)

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:

1. whether the group admits a finite presentation or not;

2. 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.)