Automorphism groups of \(q\)-trigonal planar Klein surfaces and maximal surfaces
A Klein surface \(X\) is said to be \(q\)-trigonal if it admits an automorphism \(\varphi\) of order three such that the algebraic genus of the quotient \(X/\langle \varphi \rangle\) is equal to \(q\). A Klein surface is said to be planar if it has topological genus 0 and \(k\geq 3\) boundary components. In this paper the authors give the complete list of the automorphism groups of \(q\)-trigonal planar Klein surfaces. This result is based on careful research on the list of the automorphism groups of planar Klein surfaces presented by \textit{E. Bujalance} [Manuscr. Math. 56, 105--124 (1986; Zbl 0605.14030)]. The authors also determine the \(q\)-trigonal planar Klein surfaces which have maximal symmetry, that is, the surfaces admitting the automorphism group of maximal order \(12(k-2)\). These surfaces, called maximal surfaces, are constructed by using NEC groups, and the fundamental polygons are presented by pasting the images of the Lambert quadrilateral.
scientific article; zbMATH DE number 5573652