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Proceedings of the American Mathematical Society

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Topologically unrealizable automorphisms of free groups

Author: John R. Stallings
Journal: Proc. Amer. Math. Soc. 84 (1982), 21-24
MSC: Primary 20E05; Secondary 57M99
MathSciNet review: 633269
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Abstract: Let $ \phi :F \to F$ be an automorphism of a finitely generated free group. It has been conjectured (I heard it from Peter Scott) that the fixed subgroup of $ \phi $ is always finitely generated. This is known to be so if $ \phi $ has finite order [1], or if $ \phi $ is realizable by a homeomorphism of a compact $ 2$-manifold with boundary [2]. Here we give examples of automorphisms $ \phi $, no power of which is topologically realizable on any $ 2$-manifold; perhaps the simplest is the automorphism of the free group of rank 3, given by $ \phi (x) = y$, $ \phi (y) = z$, $ \phi (z) = xy$.

References [Enhancements On Off] (What's this?)

  • [1] J. L. Dyer and G. P. Scott, Periodic automorphisms of free groups, Comm. Algebra 3 (1975), 195-201. MR 0369529 (51:5762)
  • [2] W. Jaco and P. B. Shalen, Surface homeomorphisms and periodicity, Topology 16 (1977), 347-367. MR 0464239 (57:4173)
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Article copyright: © Copyright 1982 American Mathematical Society

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