Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck’s program of ‘Dessins d’ enfants’, aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups. We would like to thank Fabio Tonoli for helping us with the pictures.