Collins, D. J.; Turner, E. C. An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element. (English) Zbl 0787.20022 J. Pure Appl. Algebra 88, No. 1-3, 43-49 (1993). The main theorem of this paper is stated in the title. According to the well-known theorem of M. Bestvina and M. Handel [Ann. Math., II. Ser. 135, No. 1, 1-51 (1992; Zbl 0757.57004)], the fixed point subgroup of an automorphism of a free group of finite rank \(n\), is also a free group of rank at most \(n\), so a fixed point subgroup of maximal rank has rank \(n\). The proof is obtained by further analysis of relative train track maps of graphs, techniques developed by Bestvina and Handel (loc. cit.). By using their analogous theorem in free products (a generalization of the free group case) [Efficient representatives of free products (1992)] the authors prove also a similar result for an automorphism of a free product with fixed point subgroups of maximal Kurosh rank. Reviewer: S.Andreadakis (Athens) Cited in 5 Documents MSC: 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 20F65 Geometric group theory Keywords:fixed point subgroup; free group of finite rank; relative train track maps; free product; Kurosh rank Citations:Zbl 0757.57004 PDFBibTeX XMLCite \textit{D. J. Collins} and \textit{E. C. Turner}, J. Pure Appl. Algebra 88, No. 1--3, 43--49 (1993; Zbl 0787.20022) Full Text: DOI References: [1] Bestvina, M.; Handel, M., Train tracks and automorphisms of free groups, Ann. of Math., 135, 1-53 (1992) · Zbl 0757.57004 [2] Collins, D. J.; Turner, E. C., Efficient representatives of automorphisms of free products (1992) · Zbl 0820.20035 [3] Seneta, E., Non-negative Matrices and Markov Chains (1981), Springer: Springer New York · Zbl 0471.60001 [4] Stallings, J., Topology of finite graphs, Invent. Math., 71, 551-565 (1983) · Zbl 0521.20013 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.