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Topology, Geometry and Algebra of Grothendieck’s Dessin

Kamalinejad, Ali | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 43101 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Shahshahani, Siavash; Shahshahani, Mehrdad
  7. Abstract:
  8. 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
  9. Keywords:
  10. Absolute Galois Group ; Dessin Denfant Theory ; Belyi Map ; Quadratic Differential ; Cartographic Group

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