Cyclic trigonal Klein surfaces (Q1324188)

A compact Riemann surface of genus \(g\) is hyperelliptic if and only if it admits an automorphism of order 2 with quotient the sphere. Thus its covering group \(K\) is characterised as a subgroup of index 2 in a Fuchsian group of signature \((0; {\overbrace {2,2, \dots, 2}^{2g + 2}})\). The Fuchsian group whose quotient by \(K\) gives the automorphism group of the surface then necessarily has a restricted form. For bordered hyperelliptic Klein surfaces, a similar characterisation in terms of NEC groups holds and the groups of automorphisms of such surfaces have been determined in \textit{E. Bujalance}, \textit{J. Etayo}, \textit{J. M. Gamboa} and \textit{G. Gromadzki}, Automorphism groups of compact bordered Klein surfaces. A combinatorial approach. (1990; Zbl 0709.14021). Cyclic trigonal Klein surfaces by definition admit an automorphism of order 3 with quotient having algebraic genus zero. In this paper, such surfaces with boundary and algebraic genus \(>4\) are characterised in terms of NEC groups and various structure theorems for NEC groups used to determine their automorphism groups, the groups being cyclic, dihedral or semidirect products of these groups with a cyclic group of order 3.