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Dedicated to Ludwig Dmitrievich Faddeev on the occasion of his 70th birthday Abstract. Grothendieck's dessins d'enfants are applied to the theory of the sixth Painleve and Gauss hypergeometric functions, two classical special functions of iso- monodromy type. It is shown that higher-order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Bely˘ i 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 Painleve functions.

Grothendieck’s dessins d’enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations