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Section 2.7 Dessins (Berke)

\begin{equation*} G_\QQ \acts (X,D) \leftrightarrow (S,f) \acts G_\QQ \end{equation*}

where (X,D) is a dessin, (S,f) is a Belyi pair.

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Subsection 2.7.1 Dessins

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Definition 2.7.1.

A dessin is a pair (X,D) where X is an oriented compact topological surface and D\subset X is a finite graph:

  1. D is connected
  2. D is bicoloured
  3. X \smallsetminus D is a disjoint union of topological disks.

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Not all of these are so important (for example 3 implies 1 (but the converse does not hold)). We can also obtain a bicoloured graph from an uncoloured graph by subdividing all edges and colouring the new vertices black and the others white.

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A single edge in a sphere is, a single edge in a torus is not.

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Permutation representation of a Dessin.

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Label the edges of a dessin \{1, \ldots, N\} then

\begin{equation*} \sigma_0(i) = \text{subsequent edge in the cycle around the white vertex of }i \end{equation*}

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as we have a positive orientation on the edges

\begin{equation*} \sigma_1(i) = \text{subsequent edge in the cycle around the black vertex of }i\text{.} \end{equation*}

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Then we define

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Definition 2.7.2.

(\sigma_0, \sigma_1) is the permutation representation pair of (X,D)\text{.}

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Say

\begin{equation*} \sigma_0 = (1 , \ldots, N_1) (N_1 + 1 , \ldots, N_2)\cdots \end{equation*}

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a product of disjoint cycles. Then each of these cycles corresponds to a white vertex, where the length of the cycle is the degree of the corresponding vertex. Same for \sigma_1 and black vertices.

\begin{equation*} \{\text{cycles appearing in the decomposition of }\sigma_0\sigma_1\} \end{equation*}
\begin{equation*} \updownarrow \end{equation*}
\begin{equation*} \{\text{faces of }D\} \end{equation*}

Prove this.

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Remark 2.7.4.

D connected implies that \langle \sigma_0, \sigma_1 \rangle is transitive on \Sigma_N\text{.} As D is bicoloured the cycles on D contain an even number of edges.

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A dessin is not a triangulation of X but

\begin{equation*} \chi (X) = \#V - \#E + \#F \end{equation*}

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proof later.

\begin{equation*} (\sigma_0, \sigma_1) \leadsto (X',D) \end{equation*}
\begin{equation*} \langle \sigma_0, \sigma_1 \rangle \subseteq \Sigma_N \end{equation*}

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is transitive.

Write \(\sigma_0\sigma_1 = \tau_1 \cdots \tau_k\text{,}\) \(\tau_i\) disjoint cycles each of length \(n_i\) with \(\sum n_i = N\text{.}\) Create \(k\) faces bounded by \(2n_1, \ldots, 2n_k\) vertices, and assign the vertices white and black colours so that the graph is bicoloured. As \(\sigma_0\sigma_1\) should jump two each time we get an identification of all edges which we then glue using \(\sigma_0\text{.}\)

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Definition 2.7.7.

We say that

\begin{equation*} (X_1, D_1) \sim (X_2, D_2) \end{equation*}

if there exists an orientation preserving homeomorphism \phi \colon X_1 \to X_2\text{,} \phi|_{D_1} \colon D_1 \xrightarrow\sim D_2\text{.}

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Subsection 2.7.2 Dessins 2 Belyi pairs

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Triangle decomposition of (X,D) \leadsto T(D) a set of triangles that cover D and intersect along edges or at vertices.

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Example 2.7.9.

Edge in the sphere, add an extra vertex \times not on the edge and get a decomposition into two triangles.

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We will label triangles by T_j^\pm as there are two for each edge, by orientation some are the same.

\begin{equation*} T(D) \leadsto f_D \colon X\to \hat\CC \end{equation*}

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Glue

\begin{equation*} f_j^? \colon T_j^? \to \overline\HH^? \end{equation*}

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for ?\in \{+,-\}\text{,} where f_j^+ = f_j^- on the intersection. Where \partial T_j \xrightarrow\sim \RR \cup\{\infty\}

\begin{equation*} \text{black} \mapsto0 \end{equation*}
\begin{equation*} \text{white} \mapsto1 \end{equation*}
\begin{equation*} \times \mapsto\infty \end{equation*}

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and we have \operatorname{Branch}(f_D) \subseteq \{0,1,\infty\}\text{.} Now \deg f_D = \#\text{edges of }D\text{,} m_v(f_D) = \deg v\text{,} f_D^{-1}(\lb 0 , 1 \rb) = D\text{.} Modify X a little bit and use some lemma to get S_D \simeq_{\text{top}} X for some Riemann surface with f_D\colon S_D \to \PP^1\text{.}

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Definition 2.7.10.

(S,f) is a Belyi pair with S compact Riemann surface and f a Belyi function on S\text{.}

\begin{equation*} (S_1,f_1) \sim (S_2, f_2) \end{equation*}

if it is an isomorphism of ramified coverings.

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So we can now go in both directions.

\begin{equation*} \{\text{Dessins}\}/\sim \end{equation*}
\begin{equation*} \updownarrow \end{equation*}
\begin{equation*} \{\text{Belyi pairs}\}/\sim \end{equation*}
\begin{equation*} (X,D) \mapsto (S_D, f_D) \end{equation*}
\begin{equation*} (S, D_f) \mapsfrom (S,f) \end{equation*}

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Now to define the Galois action

\begin{equation*} G_\QQ\acts \{\text{Dessins}\}\leftrightarrow \{\text{Belyi pairs}\} \end{equation*}
\begin{equation*} \xymatrix{ (X,D)\ar[d] \ar@{-->}[r] & (X,D)^\sigma \\ (S_D,f_D)\ar[r] & (S_D^\sigma,f_D^\sigma)^\sigma \ar[u] } \end{equation*}

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The G_\QQ action is faithful on dessins of genus g\text{.}

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Example 2.7.11.

Same example \PP^1 with a single edge, f_D = z\text{,} \deg f_D = \# edges, m_v(f)= \deg v\text{.}

String.

\(n\) star.