Symmetries of quasiplatonic Riemann surfaces (Q2633990)

A \textit{triangle group} \(\Delta=\Delta(l,m,n)\) is a subgroup of the orientation-preserving isometries of \(\mathbb{P}^1(\mathbb{C})\), \(\mathbb{C}\) or the upper half plane \(\mathbb{H}\) with the hyperbolic metric, hence a group of Möbius transformations, with canonical generators \(X\), \(Y\) and \(Z\) satisfying \(X^| =Y^m=Z^n=XYZ=1\). Each triangle group \(\Delta\) has an extension by a reflection \(T_1\) in the geodesic through the fixed points of two of canonical generators. If two of \(l\), \(m\) and \(n\) are equal, say \(l=m\), \(\Delta\) has another extension by a reflection \(T_2\) transposing the fixed points of \(X\) and \(Y\). Let \(S\) be a \textit{quasiplatonic} Riemann surface. This means that \(S\) is uniformized by a normal subgroup \(N\) of finite index in a cocompact triangle group \(\Delta\). The quasiplatonic surface \(S\) carries a regular dessin in Grothendieck's terminology. The group \(G=\Delta/N\) acts on \(S\) as a group of conformal automorphisms on \(S\). A Riemann surface is \textit{symmetric} if it possesses an anticonformal involution. \textit{D. Singerman} proved in Theorem 2 of [Math. Ann. 210, 17--32 (1974; Zbl 0272.30022)] that a quasiplatonic surface \(S\) is symmetric if \(G\) admits one of the two kinds of automorphisms which \(T_1\) and \(T_2\) give rise to. However, the existence of these automorphisms does not fully characterize symmetric quasiplatonic surfaces. In this paper the authors find two more additional conditions ((3) and (4) in Theorem 1.1) that a symmetric quasiplatonic surface must satisfy: one of the conditions is again described by the existence of a special automorphism of \(G\) and the other is that \(S\) has genus one. The authors prove also in the final section that none of the four conditions in Theorem 1.1 satisfied by some symmetric quasiplatonic surface is implied by the other ones. The results in this paper are related to properties of dessins d'enfants or hypermaps, and the authors prove them using partly the theory of hypermaps.