Automorphism groups of \(q\)-trigonal planar Klein surfaces and maximal surfaces. (English) Zbl 1171.30016

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 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.


30F50 Klein surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)


Zbl 0605.14030
Full Text: DOI Link


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