On the geometrization of the absolute Galois group

A method for exhibiting the equation of an algebraic curve for a large class of dessins on an orientable closed surface is developed, and the action of the absolute Galois group is realized as modification of the associated cell structures. An application to quadratic differentials is discussed. A Galois invariant measure on $\mathbb{CP}(1)$ is introduced. Towers of (flat) refinements of dessins are introduced that relate the inverse system structure of the absolute Galois group to geometric/combinatorial structures on surfaces.