Skip to main content

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.

Subsection 2.7.1 Dessins

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.

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.

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

Permutation representation of a Dessin.

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*}

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*}

Then we define

Definition 2.7.2.

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

Say

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

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.

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.

A dessin is not a triangulation of \(X\) but

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

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*}

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 \(\sigma_0\text{.}\)

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 \(\phi \colon X_1 \to X_2\text{,}\) \(\phi|_{D_1} \colon D_1 \xrightarrow\sim D_2\text{.}\)

Subsection 2.7.2 Dessins 2 Belyi pairs

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

Example 2.7.9.

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

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*}

Glue

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

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*}

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 \(f_D\colon S_D \to \PP^1\text{.}\)

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.

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*}

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*}

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

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.