In Katok's book, we have a defintion as following:
If $\Gamma$ is a subgroup of finite index of some $\Gamma(A,\mathcal{O})$, then we call $\Gamma$ a Fuchsian group derived from a quaternion algebra $A$.
We know that subgroup of a Fuchsian group is also a Fuchsian group. So, what is the motivation of "subgroup of finite index" in that definition?