I'm currently trying to construct simple Riemann Surfaces in the way of Hurwitz (see e.g. here):
Given the complex plane $E$, which originates at $O$ and $w$ non-identical points $a_k$. Cut the plane from $O$ to each $a_k$, which shall be denoted as the cutted complex plane $E^*$. Cutlines are called $l_k$ and have a left and right side called $l^\pm_k$. Assume we have $n$ copies of $E^*$ and they are labelled, call'em $E^*_m$. At each $l_k$ there shall be a mapping $S_k$, i.e. a permutation, for which $$S_k: \{E^*_1,E^*_2,\dots,E^*_n\} \to \{E^*_{S_k(1)},E^*_{S_k(2)},\dots,E^*_{S_k(n)}\} $$ holds. For my personal reason of interest, all my examples would further obey $w=n$. Hurwitz' construction of Riemann surfaces, asks for all permutations $S_k\in \mathbb S$ of the copies of the cutted complex planes $E^*$ at the branching points $a_k$, to fulfill: $$ S_1S_2S_3\cdots S_w=1, \tag{1} $$ which ensures that there is no ramification at the origin.
So I started to build some of these products of permutations based on ramification profile of length 3 (all permutations cycle 3 complex planes) and currently found some working examples for $n=w=3$ and $n=w=4$.
So I wonder why the order in $(1)$, the product of the elements of $\mathbb S$, seems to be of no interest. Why is that? What makes the difference from one ordering to the other, when I look at the resulting Riemann surface?
But after intense number crunching I also found a counterexample (based of Heawood's graph) with $n=w=7$.
Remark: Due to my construction based on bipartite 3-regular graphs, I always assume $n=w$. I identify black vertices with the complex planes $E^*_m$ and white vertices with the (oriented) permutations $S_k$.
So obviously a subclass of sets $\mathbb S$, i.e. bicubic graphs, fulfills certain properties, to make $(1)$ to be true...
UPDATE: I found this reference "The Riemann-Hurwitz formula for the centralizer of a pair of permutations" by Antonio Machi, making a somehow similarly unspecific statement about $(1)$ to be fulfilled in some order...