\(q\)-trigonal Klein surfaces (Q1852732)

For a given nonnegative integer \(q\), a complex Klein surface \(S\) is said to be \(q\)-trigonal if it admits an order three automorphism \(f\) such that the quotient \(S/(f)\) has algebraic genus \(q\). If \(D\) denotes the hyperbolic plane and \(\Gamma\) is a surface non-Euclidean crystallographic (NEC) group uniformizing \(S\), i.e., \(S= D/\Gamma\), the authors prove that \(S\) is \(q\)-trigonal if and only if \(\Gamma\) contains a normed NEC subgroup \(\Gamma^*\) of index three and algebraic genus \(q\). Moreover, such a subgroup \(\Gamma^*\) is unique if \(9q+ 4\) is smaller than the algebraic genus \(p\) of \(S\). In such a case it is said that \(\Gamma^*\) is the \(q\)-trigonality group of \(S\). This fact is used to determine, for fixed \(p\geq 2\), the finite set consisting of those nonnegative integers \(q\) such that there exists a \(q\)-trigonal compact Klein surface \(S\) of algebraic genus \(p\). Hence, the signature of the \(q\)-trigonality group \(\Gamma^*\) is also computed. The paper, whose structure is transparent, contains also several corollaries concerning the relationship between the topological type and the \(q\)-trigonal character of a surface. The proofs are very clear and use, essentially, the combinatorial theory of compact Klein surfaces. The authors propose some open questions concerning the automorphisms groups of \(q\)-trigonal surfaces and the study of how the fundamental region of the uniformizing NEC group of a surface reflects its \(q\)-trigonal nature.