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An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element. (English) Zbl 0787.20022

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.

MSC:

20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F65 Geometric group theory

Citations:

Zbl 0757.57004
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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
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