On classification of cyclic orientation-reversing actions of big order on closed surfaces

Abstract

At the end of 19 century, A. Wiman proved that the order of any orientation-preserving periodic self-homeomorphism of a closed orientable surface of genus g>1 does not exceed 4g+2. Later in the 1960s, W. Harvey showed that this maximum possible order is attained for each g. In the middle of the 1980s, J.J. Etayo showed that any finite cyclic group ZN generated by an orientation-reversing periodic self-homeomorphisms of a closed orientable surface Sg of genus g>1 has order bounded above by 4g+4 and 4g−4 for g even and odd respectively, and these bounds are sharp for all g. Five years later, S. Wang proved these results more directly in a purely topological way. The question to which extent the constructions of Etayo and Wang are unique was the original motivation for the present paper. Here we classify up to topological conjugation orientation-reversing actions of a cyclic group ZN on Sg, in function of a possible type of the quotient orbifold Sg/ZN, provided that N>2g−2. In particular, we prove that Etayo–Wang extremal actions are unique up to topological conjugations.