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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 43101 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Shahshahani, Siavash; Shahshahani, Mehrdad
- Abstract:
- Grothendieck’s theory of dessin d’enfant makes a connection between piecewise flat metrics and conformal structures on a compact surface R on one hand, and the defining equation for R whose coefficients lie in an algebraic number field on the other. This connection is realized by a combinatorial structure (a dessin) on a given surface R.In this thesis we begin by briefly reviewing Grothendieck’s theory of dessin d’enfant and Belyi’s theorem and also the approximation of a Belyi map via Thurston’s theory of circle packings.We then introduce a method for computing the explicit equation of the Riemann surface and Belyi map associated to a dessin. Also we explain how the absolute Galois group Gal( Q=Q) acts on Belyi pairs and we present a geometric meaning for the action of Gal( Q=Q) on these pairs.Cartographic and monodromy groups associated to a dessin are introduced and a Galois representation on equivalence classes of cartographic groups is constructed. By defining flat refinements of a dessin, geometric representations for elements of the absolute Galois group are exhibited.
Finally, the relation with quadratic differentials and a natural measure on CP(1), invariant under Gal( Q=Q), for proper dessins is introduced - Keywords:
- Absolute Galois Group ; Dessin Denfant Theory ; Belyi Map ; Quadratic Differential ; Cartographic Group
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