Grothendiecks dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations

Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Belyĭ functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painlevé functions.