Let $O$ be the ring of $S$-integers in a real quadratic number field. Let $G$ be an $S$-arithmetic subgroup of $SL_2(O)$ whose intersection with $SL_2(\mathbb Z)$ is not of finite index in $SL_2(\mathbb Z)$.

*Assume the action of $G$ is properly discontinuous for simplicity.*

Can the fundamental domain of the action of $G$ on the complex upper half plane $\mathbb H$ have finite volume?

What if we assume $G$ to be of finite index in $SL_2(O)$?

What if we remove the hypothesis that $G\cap SL_2(\mathbb Z)$ be of infinite index in $SL_2(\mathbb Z)$?

The reason I ask is simply because I wish to understand whether the "quotient" of $\mathbb H$ by $G$ could behave "well" in certain situations.