Signature of a group action on a curve (reviewed)

Let $G$ be a group of automorphisms acting on a curve $X/\C$ of genus at least $2$, let $g_0$ be the genus of the quotient $Y:=X/G$, and let $B$ be the set of branch points of the projection $\phi\colon X\to Y$. For each standard generator for $\pi_1(Y-B,y_0)$ corresponding to the elements of $B=\{y_1,\ldots,y_r\}$ and a fixed pre-image of $y_0$, there is a lift of that loop to a path in $X - \phi^{-1}(B)$ starting at the particular pre-image of $y_0$. The lifted path ends at some (possibly different) pre-image of $y_0$. Varying the particular pre-images of $y_0$ and then recording the end point corresponding to that starting point induces a permutation on those pre-images.

We define $m_1 \leq m_2 \leq \ldots \leq m_r$ to be the orders of the permutations described above, one for each of the $r$ standard generators.

The sequence of integers $[g_0; m_1, \ldots, m_r]$ is the signature of the group action.