On the automorphism group of the canonical double covering of bordered Klein surfaces with large automorphism group (Q1356287)

Let \(X\) be a Klein surface with nonempty boundary and double covering \(X^+\). The aim of this article is to get information about the group \(\Aut(X^+)\) of the conformal and anticonformal automorphisms of \(X^+\) from the group \(\Aut(X)\) of automorphisms of \(X\). Consequently, it is proved here that if \(\Aut X>6(g- 1)\), where \(g\) is the genus of \(X^+\), then \(\Aut(X^+)= \Aut(X)\times C_2\) except for a finite number of cases. This result improves a previous one by \textit{C. L. May} [Glasg. Math. J. 33, No. 1, 61-71 (1991; Zbl 0718.30032)] and \textit{E. Bujalance}, \textit{A. F. Costa}, \textit{G. Gromadzki} and \textit{D. Singerman} [Glasg. Math. J. 36, No. 3, 313-330 (1994; Zbl 0810.30035)]. The Porto-Costa new result is the best possible in this line because, in the paper under review, the existence of infinitely many Klein surfaces \(X\) with different topological type satisfying \(\#\Aut(X)= 6(g- 1)\) and \(\Aut(X^+)\neq \Aut(X)\times C_2\), is established. This fact is a nice consequence of the theory of irreflexible maps on surfaces. In particular, for \(g=2\), families of surfaces with \(\#\Aut(X)= 6\) and \(\Aut(X^+)\neq \Aut(X)\times C_2\) were constructed recently by \textit{J. Cirre} in his Ph.D. thesis in 1997. Finally, the authors conclude that the existence of an automorphism of \(X\) with fixed points and prime order \(p\) implies, provided the order \(p\) is large enough, that \(\Aut(X^+)\neq \Aut(X)\times C_2\) with a unique exception for each value of \(g\).