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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.

MSC:

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

Citations:

Zbl 0605.14030
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References:

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