Schottky double covers (Q2269959)

\textit{H. M. Farkas} [J. Anal. Math. 30, 150-155 (1976; Zbl 0348.32006)] proved that for a two-sheeted unramified covering \(\tilde S\) of a hyperelliptic Riemann surface \(S\), the hyperelliptic involution of \(S\) lifts to a conformal involution of \(\tilde S\). In the paper under review, the author proves a similar result for Schottky groups: For a hyperelliptic Riemann surface \(S\) and its unramified double cover \(\tilde S\), there exists a hyperelliptic Schottky group \(G\) uniformising \(S\) and containing an index-2 subgroup \(H\) uniformising \(\tilde S\) such that the hyperelliptic involution lifts to elliptic transformations which normalize \(H\).