Let $\Gamma \subseteq PSL\_2(\mathbb{R})$ be a Fuchsian group, possibly containing elliptic elements. Is it true that $N(\Gamma) / \Gamma$, where $N(\Gamma)$ the normalizer of $\Gamma$ in $PSL\_2(\mathbb{R})$, is isomorphic to the automorphism group of $\Gamma \backslash \mathcal{H}^\*$?


Here, $\mathcal{H}^\*$ is the union of the upper-half plane and the set of cusps of $\Gamma$.


If so, can you point me to a reference, please?



Fuchsian groups and automorphisms of Riemann surfaces
The compact Riemann surface $\Gamma\backslash\mathcal H^\*$ may have authomorphisms that do not preserve the set of points corresponding to the cusps of $\Gamma$, so I believe the answer is no.


