# Criterion for a subgroup of $PSL_2(\mathbb R)$ to be Fuchsian

Let $\Gamma$ be a finitely generated subgroup of $PSL_2(\mathbb R)$.

I'm looking for effective criteria (idealy necessary and sufficient but just necessary would be a good start) ensuring that $\Gamma$ is

1.- Fuchsian (ie. $\Gamma$ is a discrete subgroup); or

2.- Fuchsian of the first type (ie. the Riemann surface $\mathbb H/\Gamma$ is of finite hyperbolic area); or

3.- Fuchsian and co-compact (ie. the Riemann surface $\mathbb H/\Gamma$ is compact).

I'm not familiar with this field, but I know that there are a lot of works on this question, especially when $\Gamma$ is generated by two elements. In this case, one can certainly find some satisfying criteria or algorithms in Gilman's memoir `Two-generators discrete subgroups of $PSL_2(\mathbb R)$'.

Main question: what about the case when $\Gamma$ is generated by more than two elements?

I think that $\Gamma$ is Fuchsian iff every subgroup generated by 2 elements of $\Gamma$ is discrete (is it true? A confirmation and better, a reference, would be welcome). Anyway, I don't see how this could be used in an effective manner to verify that a given $\Gamma$ is discrete or not...

Main question (made more precise): assume that $\Gamma$ is generated by $n\geq 3$ elements. Is there a criterion involving the $n$ generators of $\Gamma$ all together that must be verified if this group is Fuchsian (or Fuchsian of the first type, or Fuchsian and cocompact)?

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Secondary question: in case of two-generators subgroups, are the existing criteria/algorithms about the properties 1, 2 and 3 implemented? If yes, where is it possible to find softwares/codes allowing to make explicit computations?