On \(pq\)-hyperelliptic Klein surfaces (Q946868)

A compact Klein surface \(X\) is a compact surface, orientable or non-orientable, with \( k\geq 0\) boundary components. If \(X\) is orientable and \(k=0\), then \(X\) is a Riemann surface. The algebraic genus \(d\geq2\) of \(X\) is the genus of the Riemann surface \(X^+\) that is its complex double. The algebraic genus \( d=\eta g+k-1\), where \(g\) is the topological genus of \(X\) and \(\eta=2\) or \( \eta=1\) according to \(X\) be orientable or not. Klein surfaces are uniformized by means of non-Euclidean crystallographic groups (NEC groups, in short). Given a Klein surface \(X\) of algebraic genus \(d\geq2\), \(X\) can be represented as \(\mathcal{H}/\Gamma\) for some NEC group \(\Gamma\), where \(\mathcal{ H}\) denotes the hyperbolic plane. A Klein surface is said to be \(p\)-hyperelliptic if there exists an involution \(\rho\) such that the quotient \(X/\rho\) has algebraic genus \(p\). In the paper under review \(pq\)-hyperelliptic Klein surfaces, surfaces which are simultaneously \(p\)- and \(q \)-hyperelliptic, are studied. It is proved that given integers \(0\leq p\leq q\), \(q\neq 0\) then there exists a \(pq\)-hyperelliptic Klein surface of algebraic genus \(d\) if and only if \(2q-1\leq d\leq 2p+2q+1\). This result generalizes for Klein surfaces a similar result for Riemann surfaces from the second author in [\textit{E. Tyszkowska}, Colloq. Math. 103, No. 1, 115--120 (2005; Zbl 1080.30037)]. The authors study the product of \(p\)- and \(q\)- involutions obtaining some results about conditions such that these involutions commute. The last section of the paper is devoted to the study of the topological types of Klein surfaces with commuting \(p\)- and \(q\)- involutions. It is proved that with few exceptions a \(pq\)-hyperelliptic Klein surface can have arbitrary number \(k\) of boundary components \(0\leq k\leq d+1\). The authors start from a Klein surface \(X\) of genus \(d\) with maximal number of components. Let \(\Lambda \) be an NEC group of signature \((0;+;[2^r];\{(2^s),(-)^l\})\) where the numbers \(d,r,s\) and \(l \) must satisfy some equalities. Let \(\theta :\Lambda \rightarrow G=\langle\delta ,\rho\rangle\) an epimorphism defining a Klein surface having algebraic genus \(d\), \(k\) boundary components and two \(p\)- and \(q\)-involutions, \(\delta \) and \( \rho \) respectively, whose product is a \(t\)-involution. By means of four types of modifications of the signature of \(\Lambda \) or of the definition of the epimorphism \(\theta \) the authors obtain different numerical conditions which \(p,q,t\) and \(d\) must satisfy. It is done in five theorems. Some results about proper periods and cycle-periods of normal NEC subgroups from [\textit{E. Bujalance} et al., Lect. Notes Math. 1439 (1990; Zbl 0709.14021)] play an important technical role in this paper.