Maximum Order of Periodic Outer Automorphisms of a Free Group

Let Fn be a free group with rank n, and denote by OutFn its outer automorphism group. For arbitrary n, consider the orders of periodic elements in OutFn or, equivalently, the orders of finite cyclic subgroups of OutFn. By considering group actions on finite connected graphs, we obtained the number-theoretical characterization of these orders. Comparing the results with those for cyclic subgroups of finite symmetric groups asymptotic estimation for the maximum order cn is derived.