Maximum order of periodic outer automorphisms of a free group
Landau proved that the maximum order of an element in the symmetric group \(\Sigma_n\) behaves asymptotically like \(\exp(\sqrt{n\log n})\). In the present paper, it is shown that this Landau type estimate holds also for the maximum order of torsion elements in the outer automorphism group \(\text{Out}F_n\) (automorphisms modulo inner automorphisms) of the free group \(F_n\) of rank \(n\). The proof consists of two parts. First, a complete number theoretical description of the possible orders of elements in \(\text{Out}F_n\) is given, by a careful combinatorial analysis of finite cyclic group actions on finite graphs (on the basis of the result that any finite subgroup of \(\text{Out}F_n\) can be induced by an action of the group on a finite graph with fundamental group \(F_n\)); this characterization is also obtained in a paper of \textit{D. G. Khramtsov} [Algebra Logika 26, No. 3, 376-394 (1987; Zbl 0657.20026)], by different methods. The main result of the present paper is then deduced from this characterization using some number theoretical considerations and the above Landau estimate for the maximal orders in the symmetric groups. In a recent paper of \textit{G. Levitt} and \textit{J.-L. Nicolas} [J. Algebra 208, No. 2, 630-642 (1998; Zbl 0930.11070)], the same Landau type estimate is also obtained for the maximum orders of torsion elements in the general linear groups \(\text{GL}(n,\mathbb{Z})\). (Also submitted to MR).
scientific article; zbMATH DE number 1428249
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