On two recent geometrical characterizations of hyperellipticity (Q1884574)

The authors introduce the notions of prehyperelliptic and central hyperelliptic fundamental polygons for a surface Fuchsian group \(\Gamma \), and they prove the following Theorem. Let \(\Gamma\) be a surface Fuchsian group which uniformizes the compact, genus \(g\geq 2\), Riemann surface \(X\). Then, the following statements are equivalent: (i) \(X\) is hyperelliptic; (ii) \(\Gamma\) admits a prehyperelliptic and fundamental polygon having a rotational symmetry of order 2; (iii) \(\Gamma\) admits a central hyperelliptic fundamental polygon. As a consequence, the authors get new and very beautiful proofs of two recent theorems by \textit{B. Maskit} [A new characterization of hyperellipticity, Mich. Math. J. 47, No. 1, 3--14 (2000; Zbl 0959.30025)] and \textit{P. Schmutz Schaller} [Geometric characterization of hyperelliptic Riemann surfaces, Ann. Acad. Sci. Fenn., Math. 25, 85--90 (2000; Zbl 0951.30036)]. The first states that \(X\) is hyperelliptic if and only if it has an evenly spaced geodesic necklace. The second characterizes hyperelliptic genus \(g\) surfaces as those admitting a \((2g-2)\)-star. The whole paper, whose statements and proofs are transparent, emanates a genuine geometrical flavor.