# Fuchsian groups of singly branched covers

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 $$\Gamma_X, \Gamma_1, \Gamma_2$$?