Lifting the hyperelliptic involution of a Klein surface
- 45 Downloads
We consider unbranched normal coverings \(X\rightarrow X'\) between compact Klein surfaces of algebraic genus bigger than one where \(X'\) is hyperelliptic. Here unbranched means that the fixed point set of the group G of covering transformations is either empty or projects onto the boundary of \(X'\). We find a criterion which determines whether the hyperelliptic involution of \(X'\) lifts to an automorphism of X. The study splits naturally into six cases according to the different topological types that \(X'\) may possess. Our results apply nicely to the case when G is abelian, showing for instance that if G has odd order then the hyperelliptic involution always lifts. We also apply the criterion to particular presentations of other types of groups.
KeywordsKlein surface Normal covering Automorphism group Lifting automorphisms
Mathematics Subject Classification30F50 20H10
The authors wish to thank the referees for their suggestions and comments which have contributed to get a more readable paper.
- 4.Bujalance, E., Bujalance, J.A., Gromadzki, G., Martinez, E.: The groups of automorphisms of nonorientable hyperelliptic Klein surfaces without boundary. In: Proceeding of “Groups-Korea (Pusan, 1988),” Lecture Notes in Mathematics, vol. 1398, pp. 43–51. Springer, Berlin (1989)Google Scholar
- 9.Farkas, H.M.: Unramified coverings of hyperelliptic Riemann surfaces. Complex Analysis, I (College Park, Md., 1985–1986), Lecture Notes in Mathematics, vol. 1275, pp. 113–130. Springer, Berlin (1987)Google Scholar
- 12.Howie, J., Thomas, R.M.: Proving certain groups infinite. In: Geometric Group Theory, vol. 1 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 181, pp. 126–131. Cambridge University Press, Cambridge (1993)Google Scholar
- 15.Natanzon, S.M.: Klein surfaces, (Russian) Uspekhi Mat. Nauk 45 (1990), no. 6(276), 47–90, 189; translation in Russian Math. Surveys 45 (1990), no. 6, 53–108Google Scholar