Let $X/\mathbb{C}$ be an algebraic curve with genus $g \geq 2$. Then by the uniformization theorem, with $X(\mathbb{C})$ viewed as a Riemann surface, it can be realized as the quotient $\mathbb{H}/\Gamma_X$ where $\mathbb{H}$ is the upper half-plane and $\Gamma_X$ a discrete subgroup of $\text{PSL}_2(\mathbb{R})$.

Now suppose that $Y_{P_1}$ is a singly branched cover of $X$ of degree $n$, branched precisely at $P_1 \in X(\mathbb{C})$. $Y_{P_1}$ has genus $g' > 1$, where $g'$ is an explicit function of $g$ and $n$. Now suppose that $P_1 \ne P_2$ is another point on $X(\mathbb{C})$. An argument of Fulton (https://www.jstor.org/stable/1970748?seq=1, 1.3) shows that if $Y_{P_2}$ is a singly branched cover of $X$ branched precisely at $P_2$ of degree $n$, then $Y_{P_1}, Y_{P_2}$ can be topologically identified. In particular, $Y_{P_2}$ and $Y_{P_1}$ will have the same genus.

My question is, suppose that $Y_{P_1} \cong \mathbb{H}/\Gamma_1$ and $Y_{P_2} \cong \mathbb{H}/\Gamma_2$, for two Fuchsian groups $\Gamma_1, \Gamma_2$. What is the relationship between $\Gamma_X, \Gamma_1, \Gamma_2$?


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