Extensions of finite cyclic group actions on non-orientable surfaces (Q2847035)

Let \(S\) be a Riemann surface, and \(\mathrm{Aut}(S)\) the group of automorphisms of \(S\). The question of whether a given cyclic group acting on \(S\) is the full group \(\mathrm{Aut}(S)\), was considered by \textit{E. Bujalance} and \textit{M. D. E. Conder} [J. Lond. Math. Soc., II. Ser. 59, No. 2, 573--584 (1999; Zbl 0922.20054)]. In this article, the authors generalize this question to non-orientable Klein surfaces.NEWLINENEWLINEUsing the combinatorial theory of non-euclidean crystallographic groups (NEC groups), the authors perform a case-by-case analysis of the 15 so-called \textit{normal pairs} of NEC signatures, which correspond to pairs \((\sigma, \sigma')\) such that every NEC group \(\Gamma\) with signature \(\sigma\) is a normal proper subgroup of another NEC group \(\Gamma'\) with signature \(\sigma'\).NEWLINENEWLINEThe main theorem of this paper states that any action on \(S\) of a cyclic group \(C_n\) with non-maximal signature extends to an action of a larger group. This contrasts with the orientable case where such actions do not always extend. The authors also give the complete table of the largest possible order of a cyclic group acting on a non-orientable surface of genus \(g\) and the possible signatures for such a surface. Similarly, for a given \(n\), they determine the smallest algebraic genus of a non-orientable surface on which \(C_n\) acts as the full group \(\mathrm{Aut}(S)\).