Automorphism groups of compact bordered Klein surfaces with invariant subsets (Q996057)

Let \(S\) be a compact bordered Klein surface of algebraic genus \(g\geq 2\) and \(G\) be a group of its automorphisms. A set \(B\subset S\) is called invariant under the action of \(G\) if \(G(B)=B\). Upper bounds for the order of \(G\) are obtained under additional condition that \(G\) has one or two invariant subsets of fixed cardinals. It is proved that \(| G| \leq 4(g-1)+4| B| \) if \(B\) is a non-empty invariant subset (Theorem~1); \(| G| \leq 2(g-1)+2| B| +2| C| \) for two different non-empty invariant subsets \(B\) and \(C\). The estimates are sharp. In the last section the authors consider partitions \({\mathcal B}=\bigcup_{j=1}^m {\mathcal B}_j\) of the set of boundary components of \(\partial S\) such that each \({\mathcal B}_j\) is invariant under the action of \(G\). In Theorem~4 it is proved that, with the exception of a few particular cases, \(| G| \leq 4(g-1)\) for \(m=2\) and \(| G| \leq 2(g-1)\) for \(m=3\). The main tool in the proofs is the generalized Hurwitz ramification formula for Klein surfaces obtained by \textit{C.~L.~May} [Pac.~J. Math. 59, 199--210 (1975; Zbl 0422.30037)].