Can anyone help to translate the french wiki on the topology of dessin d'enfants:
Soit ${\displaystyle S}$ la sphère privée de trois points...
Merci beaucoup...
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It seems to mean "let $S$ denote the sphere with three points deleted". – fred goodman Mar 25 '18 at 20:06
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@fredgoodman thanks, can you do the (linked) rest as well? – draks ... Mar 25 '18 at 20:07
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No sorry, no time at the moment. – fred goodman Mar 25 '18 at 20:08
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1But a bit of general advice: you can learn to read mathematics in French rather quickly by reading something on a subject that you know and looking up lots of words. Within a few days, you will be able to read rather well. – fred goodman Mar 25 '18 at 20:10
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Let $S$ be the sphere with points $P_1, P_2$, $P_3$ deleted. For a fixed basepoint $b\in S$, the topological fundamental group $\pi_1^{top} = \pi_1^{top}(S, b)$ is free on two elements. More precisely, the simple loops $x$ and $y$ around $P_1$ and $P_2$ are canonical generators of $\pi_1^{top}$, and the simple loop $z$ around $P_3$ has $xyz = 1$. Recall that there exists a bijection between the set of isomorphism classes of finite covers of $S$ and conjugation classes of finite-index subgroups of $\pi_1^{top}$.
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With a dessin d'enfant given, it's now possible to define a right action by $\pi_1^{top}$ on its edges: $x$ (resp. $y$) sends an edge $e$ to the first edge obtained by turning anticlockwise from the black (resp. white) vertex of $e$. This action, described in Figure 3, is called the monodromy action;
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it's transitive because the underlying graph of the dessin d'enfant is assumed to be connected. In particular, the stabilizers of the edges are conjugate subgroups of $\pi_1^{top}$ of index equal to the degree of the dessin d'enfant. We thus associate to a dessin d'enfant a conjugation class of subgroups of $\pi_1^{top}$. Conversely, given such a conjugation class $C$, it's possible to associate a dessin d'enfant to it: its edges are by definition the elements of the group $\Gamma = \pi_1^{top}/H$ of right cosets of a representative $H$ of $C$. This right action by an element of $\pi_1^{top}$ defines an action on $\Gamma$, and two edges have a black (resp. white) vertex in common iff they belong to the same orbit under the action of $\pi_1^{top}$ generated by $x$ (resp. $y$). It's easily verified that these two constructions are inverse of each other. Thus:
Proposition/Construction: There exists a bijection between dessins d'enfants and isomorphism classes of finite topological covers of the sphere with three points deleted.
The following paragraph gives a more visual and intuitive description of this correspondence.
anomaly
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merci beacoup encore une fois. Hope you don't mind my figure addings... – draks ... Mar 25 '18 at 21:59
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Do you know if $H$ in $\Gamma=\pi_1^{top}/H$ is the upper half plane? And what does the $top$-flag at the fundamental group $\pi_1$ mean? – draks ... Mar 26 '18 at 18:09
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1$H$ is a subgroup of $\pi_1^{top}$ modulo conjugation. $\Gamma$ is a group, not a topological space. The 'top' tag is just there to distinguish it from some of the more exotic constructions in algebraic geometry that are also called fundamental groups. – anomaly Mar 26 '18 at 19:35