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.


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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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