On automorphisms of Klein surfaces with invariant subsets

Abstract

It is well known that a group of automorphisms $G$ of an unbordered Klein surface $X$ of topological genus $g\geq 2$ in the orientable case and $g\geq 3$ otherwise has at most $84(g-\varepsilon)$ elements, where $\varepsilon =1$ or $2$ respectively. In the middle of the fifties, Oikawa used the cardinality $k$ of a $G$-invariant subset to introduce the bound $\lvert G\rvert \leq 12(g-1)+6k$ in the orientable case. Much later, T. Arakawa has generalized this result, involving $s=2$ or $3$ such subsets and showing in addition that the bound for $s=3$ is sharp for infinitely many configurations. Here we improve the bound of Arakawa for $s=2$, showing in particular that the last is never attained. In both orientable and non-orientable case, we also find bounds for arbitrary $s$ and show their sharpness for infinitely many topological configurations. Using another well known theorem of Oikawa and the canonical Riemann double cover, we get similar results for bordered Klein surfaces.