Conjugators of Fuchsian groups and quasiplatonic surfaces

Abstract

Let G be a Fuchsian group containing two torsion free subgroups defining isomorphic Riemann surfaces. Then these surface subgroups K and αKα −1 are conjugate in PSL(2, R), but in general the conjugating element α 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 α 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.

How many quasiplatonic surfaces?

“…The lower bound follows similarly, but a quasiplatonic surface can be obtained by up to seven different types of regular dessins ( [2]), and another overcount can happen: in a fixed triangle group = (p, q, r) several torsion free normal subgroups can be PSL 2 (R)-conjugate, leading to isomorphic surfaces. In [3,Theorems 5,6,7], it is shown that such conjugations take place in a finite extension of which is again a triangle group. By Singerman's work [8] the maximal possible index between Fuchsian triangle groups is known to be 24 (occurring for (2, 3, 7) ⊃ (7, 7, 7)), so we have at most 168 normal subgroups counted in the proof of Theorem 1 leading to isomorphic surfaces (by a more detailed analysis, this number can considerably decreased).…”

“…2) Counting regular dessins in genera 0 and 1 is different from higher genera. In genus 0 the Riemann sphere is the only surface, however having an infinity of regular dessins defined by the cyclic and dihedral triangle groups of signatures (1, n, n), (2, 2, n) and those corresponding to the platonic bodies, i.e., (2,3,3), (2,3,4), (2,3,5).…”

“…In genus 1 the triangle groups of signatures (3,3,3), (2,3,6), (2,4,4) have infinitely many torsion free normal subgroups of finite index, acting by translations on the complex plane. As quotients one obtains infinitely many non-isomorphic elliptic curves with regular dessins, but all fall in two isogeny classes only (see [10]) having complex multiplication by the fields of third or forth roots of unity, respectively.…”

Fields of definition of uniform dessins on quasiplatonic surfaces

“…One can in fact show [11] that regular dessins -those for which the group of colourand orientation-preserving automorphisms acts transitively on the edges -on surfaces of genus g > 1 are almost uniquely determined: their surfaces C are called quasiplatonic, and they can be characterized by the fact that their surface groups Γ are normal subgroups of triangle groups ∆ . It is nontrivial but not too surprising that the existence of several non-isomorphic regular dessins on the same curve C is always induced by the finitely many and well-known inclusion relations between different triangle groups.…”

Shimura curves with many uniform dessins

“…The quotient Δ/Γ acts as a group of biholomorphic automorphisms on the surface and transitively on the edges of the dessin, these edges corresponding to the residue classes of Δ mod Γ . As shown in [3] and [5], quasiplatonic surfaces can have finitely many regular dessins only, and these regular dessins are related to each other by conjugations in triangle groups and inclusion relations between them.…”