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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Topologically unrealizable automorphisms of free groups


Author: John R. Stallings
Journal: Proc. Amer. Math. Soc. 84 (1982), 21-24
MSC: Primary 20E05; Secondary 57M99
DOI: https://doi.org/10.1090/S0002-9939-1982-0633269-8
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$.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0633269-8
Article copyright: © Copyright 1982 American Mathematical Society