REUF4: Research Experiences for Undergraduate Faculty

During June 4-8, 2012, I lead a research group on the subject at the Research Experiences for Undergraduate Faculty (REUF) at ICERM in Providence, Rhode Island. Here is a summary of the main results we found during the 2012 summer program.

1. Every Belyĭ map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ of degree $\deg(\beta) = 1$ is in the form $\beta(z) = \dfrac {a \, z + b}{c \, z + d}$ where $a \, d - b \, c \neq 0$.

2. Up to fractional linear transformation, every Belyĭ map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ of degree $\deg(\beta) = 2$ is in the form $\beta(z) = \left( \dfrac {a \, z + b}{c \, z + d} \right)^2$ where $a \, d - b \, c \neq 0$.

3. Consider four distinct complex numbers $z^{(-1)}$, $z^{(0)}$, $z^{(+1)}$, and $z^{(\infty)}$ with cross-ratio $\bigl( z^{(-1)}, \, z^{(0)}; \, z^{(+1)}, \, z^{(\infty)} \bigr) = -1$ and define the rational function $\beta(z) = \left[ \dfrac {2 \, \bigl( z^{(0)} - z^{(1)} \bigr) \, \bigl( z^{(\infty)} - z^{(1)} \bigr) \, \bigl( z - z^{(0)} \bigr) \, \bigl( z - z^{(\infty)} \bigr)}{ \bigl( z^{(0)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(\infty)} \bigr)^2 + \bigl( z^{(\infty)} - z^{(1)} \bigr)^2 \, \bigl(z - z^{(0)} \bigr)^2} \right]^2$. Then $\beta(z)$ is a Belyĭ map whose associated Dessin d’Enfant $K_{2,2}$ has vertices $B = \bigl \{ z^{(0)}, \, z^{(\infty)} \bigr \}$ and $W = \bigl \{ z^{(-1)}, \, z^{(+1)} \bigr \}$.

4. Every planar complete bipartite graph $K_{m,n}$ be realized as the Dessin d’Enfant of some Belyĭ map, namely either $\beta(z) = z^n$ or $\beta(z) = 4 \, z^n/(z^n + 1)^2$.

5. Every path graph be realized as the Dessin d’Enfant of some Belyĭ map, namely $\beta(z) = \bigl( 1 + \cos \, ( n \, \arccos z) \bigr)/2$.

6. Every bipartite cycle graph be realized as the Dessin d’Enfant of some Belyĭ map, namely $\beta(z) = (z^n + 1)^2/ (4 \, z^n)$.

7. The Möbius Transformations $r(z) = (z-1)/z$ and $s(z) = z/(z-1)$ generate a subgroup of $\text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$ isomorphic to $S_3 = \left \langle r, \, s \, \bigl | \, r^3 = s^2 = (s \, r)^2 = 1 \right \rangle$.

8. Let $\phi(w)$ be a rational function. The composition $\phi \circ \beta$ is a Belyĭ map for every Belyĭ map $\beta$ if and only if $\phi$ is a Belyĭ map which maps the set $\bigl \{ (0:1), \, (1:1), \, (1:0) \bigr \}$ to itself.

9. Let $\Gamma = \bigl( B \cup W, \, E \bigr)$ be the Dessin d’Enfant associated to a Belyĭ map $\beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$. For each $\gamma(z) \in S_3$, let $\Gamma_\gamma = \bigl( B_\gamma \cup W_\gamma, \, E_\gamma \bigr)$ be the Dessin d’Enfant associated to the composition $\gamma^{-1} \circ \beta: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$. That is, $B_\gamma = \beta^{-1}\bigl( \gamma(0) \bigr)$, $W_\gamma = \beta^{-1} \bigl( \gamma(1) \bigr)$, and $E_\gamma = \beta^{-1} \bigl( \gamma([0,1]) \bigr)$.

1. If $\gamma = 1$, then $\Gamma_1 = \Gamma$ is the original Dessin d’Enfant.

2. If $\gamma = s$, then $\Gamma_s$ can be obtained from $\Gamma$ by interchanging the white vertices $W$ with the midpoints of the faces $F$.

3. If $\gamma = s \, r$, then $\Gamma_s$ can be obtained from $\Gamma$ by interchanging the black vertices $B$ with the white vertices $W$. In other words, $\Gamma_s$ is the dual graph to $\Gamma$.

4. If $\gamma = r \, s$, then $\Gamma_s$ can be obtained from $\Gamma$ by interchanging the black vertices $B$ with the midpoints of the faces $F$.

5. If $\gamma = r$, then $\Gamma_r$ can be obtained from $\Gamma$ by cyclically rotating the black vertices $B$ to the midpoints of the faces $F$ to the white vertices $W$.

6. If $\gamma = r^2$, then $\Gamma_{r^2}$ can be obtained from $\Gamma$ by cyclically rotating the black vertices $B$ to the white vertices $W$ to the midpoints of the faces $F$.