On Singerman symmetries of a class of Belyi Riemann surfaces (Q1025064)

Let \(X\) denote a compact Riemann surface of genus \(g\geq 2\). A symmetry \(\sigma\) of \(X\) is an antiholomorphic involution of \(X\). If \(X\) admits a symmetry, then we call \(X\) symmetric. The fixed point set of a symmetry \(\sigma\) consists of simple closed curves called the ovals of \(\sigma\). A group \(G\) of conformal automorphisms of \(X\) is called large if \(X/G\) has genus zero and the map \(\pi : X \rightarrow X/G\) is branched over precisely three points. In the paper under review, the author provides formulas for the number of ovals of a symmetry \(\sigma\) of a compact Riemann surface \(X\) which admits a large group of automorphisms \(G\). The author proves the results using non-Euclidean crystallographic (NEC) groups and uniformization. Suppose that \(G\) is a large group of automorphisms of \(X\). In \textit{D. Singerman}, [Math. Ann. 210, 17--32 (1974; Zbl 0272.30022)] necessary and sufficient algebraic conditions on a generating set for \(G\) were provided for \(X\) to be symmetric. This result together with a formula for the number of ovals of a symmetry from \textit{G. Gromadzki} [J. Pure Appl. Algebra 121, No.3, 253--269 (1997; Zbl 0885.14026)] which is adapted to the special case where \(G\) is a large automorphism group, allows the author to explicitly calculate the number of ovals of a symmetry of \(X\). The paper is well written.