Reducibility and orders of periodic automorphisms of surfaces (Q1197481)

An orientation-preserving homeomorphism (automorphism) of a closed orientable surface \(\Sigma_ g\) of genus \(g\) is called reducible if it leaves invariant a collection of disjoint nontrivial simple closed curves, otherwise irreducible. It is shown that the order of a periodic irreducible homeomorphism is bounded below by \(2g+1\), and that the order of a periodic reducible homeomorphism is bounded above by \(2g+2\), and \(2g\) if \(g\) is odd. Moreover all bounds are best possible. The first one is a direct consequence of the Riemann-Hurwitz formula and the well-known fact that the quotient of an irreducible action is the 2-sphere with 3 branch points (the quotient of a triangle group). For the second one, for a given order \(N\) the minimal genus of \(g\) of a surface \(\Sigma_ g\) is first determined which admits a periodic reducible homeomorphism of that order, in terms of the prime decomposition of \(N\); here lies the main work of the paper.