Periodic automorphisms of surfaces: Invariant circles and maximal orders

W. H. Meeks has asked the following question: For what g does every (orient ation preserving) periodic automorphism of a closed orientable surface of g enus g have an invariant circle! A variant of this question due to R. D. Ed wards asks for the existence of invariant essential circles. Using a constr uction of Meeks we show that the answer to his question is negative for all but 43 Values of g less than or equal to 10000, ail of which lie below g = 105. We then show that the work of S. C. Wang on Edwards' question general izes to nonorientable surfaces and automorphisms of odd order. Motivated by this, we ask for the maximal odd order of a periodic automorphism of a giv en nonorientable surface. We obtain a fairly complete answer to this questi on and also observe an amusing relation between this order and Fermat prime s.