Dear all,

my question is: Suppose we have a Fuchsian group G and another group H, containing G with finite index. Must H be a Fuchsian group? Can you give a proof? Or do you know any counter examples?

Best Ali K.

[Edit: Ali K's answer below indicates that he is interested in the case in which $H$ is also a subgroup of $\operatorname{PSL}_2(\mathbb{R})$, so I have answered the question with that additional hypothesis. However the question as stated, though it may not be the OP's intent, is probably more interesting, so I do *not* wish to officially change it. -- PLC]