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Galois actions on Dessins D’Enfants Burggraf, David S.

Abstract

Grothendieck's theory of Dessins d'Enfants is an investigation of the many connections between permutations, maps on Riemann surfaces, algebraic curves and Galois groups. Some of these connections were studied long ago by Hamilton [Ham67] and Klein [Kle56], and some are quite new and continue to be analyzed today by researchers in various mathematical fields ranging from number theory to conformal field theory. One topic of great interest that emerges is the surprising fact, which was noticed by Grothendieck, that the absolute Galois group Gal(Q/Q) acts faithfully on certain combinatorial objects known as Dessins d'Enfants. The first conference on Grothendieck's theory of Dessins d'Enfants took place at Luminy in April 1993 where many of Grothendieck's ideas were explored and shared [Sch94a]. A paper in the conference proceedings [Sch94b], shows that Gal(Q/Q) acts faithfully on Dessins d'Enfants with genus 1 and later in the same paper, another proof shows that Gal(Q/Q) acts faithfully on plane trees in genus 0. Faithful actions of Gal(Q/Q) on such simple objects motivated a systematic study of the relationships between plane trees, polynomials and Galois groups, where computer algebra systems were often used for the calculations [Sha90], [And94], [Sch94b], [Cou97], [Zvo98], [MagOO]. Work in this direction, however, has not lead to many simply defined, explicit classes of Dessins, especially where more complicated Galois groups are acting. This lead to the concluding comment by Gareth Jones in [Jon97]: "It would be interesting to produce further classes of Dessins on which Gal(Q/Q) induces more complicated groups, such as nonsolvable groups or solvable groups with unbounded derived length". One of the main aims of this thesis is to address this comment.

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