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How many quasiplatonic surfaces?
(2004)

JanChristoph SchlagePuchta
Jürgen Wolfart
 We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus <= g has o growth of typ g exp (log g). The number of nonisomorphic regular dessins of genus <= g has the same growth type.

Conjugators of Fuchsian groups and quasiplatonic surfaces
(2004)

Ernesto Girondo
Jürgen Wolfart
 Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and alphaKalpha exp(1) are conjugate in PSl(2,R), but in general the conjugating element alpha cannot be taken in G or a finite index Fuchsian extension of G. We will show that in the case of a normal inclusion in a triangle group G these alpha can be chosen in some triangle group extending G. It turns out that the method leading to this result allows also to answer the question how many different regular dessins of the same type can exist on a given quasiplatonic Riemann surface.