\(ZS\)-metacyclic groups of automorphisms of compact Riemann surfaces (Q1355675)

The authors set out to solve the minimum genus problem for a \(ZS\)-metacyclic group \(M\) of order \(pq\) where \(p\) and \(q\) are primes, \(q\mid p-1\) and \(M\) has a presentation \(\langle a,b:a^p=b^q=1\), \(b^{-1}ab=a^r\), \(((r-1)q,p)=1\), \(r^q\equiv 1\pmod p\), \(r\neq 0\rangle\). After that they establish a set of necessary and sufficient conditions on the genus and periods of a Fuchsian group \(\Gamma\) for the existence of a smooth epimorphism of \(\Gamma\) onto \(M\). Finally, they prove that the minimum genus \(q\) of a compact Riemann surface having \(M\) as its automorphism group, and the corresponding signature of a Fuchsian group \(\Gamma\) having \(M\) as a smooth quotient, are \[ \begin{aligned} q=1+\textstyle{\frac{pq}{2}\bigl(1-\frac1p-\frac2p}\bigr)\Delta(p,q,q) &\quad\text{for }q\neq 0,\tag{a}\\ q=(p-1)\Delta(2,2,p,p) &\quad\text{for }q=2.\tag{b}\end{aligned} \] {}.