Endomorphisms of free groups that preserve automorphic orbits. (English) Zbl 1013.20025
Using classical tools from combinatorial group theory, namely Nielsen transformations and the standard Whitehead graph, it is proved that if an endomorphism, \(\varphi\), of a free group of finite rank preserves a nontrivial automorphic orbit, then \(\varphi\) is actually an isomorphism, which confirms a conjecture of V. Shpilrain [Arch. Math 62, No. 5, 385-392 (1994; Zbl 0802.20024)].
Reviewer: Brigitte Servatius (Worcester)
Keywords:
finitely generated free groups; Nielsen transformations; standard Whitehead graphs; endomorphisms; automorphismsCitations:
Zbl 0802.20024References:
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| [2] | D. Lee, Primitivity preserving endomorphisms of free groups, Comm. Algebra, to appear.; D. Lee, Primitivity preserving endomorphisms of free groups, Comm. Algebra, to appear. · Zbl 1001.20014 |
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| [4] | Shpilrain, V., Recognizing automorphisms of the free groups, Arch. Math., 62, 385-392 (1994) · Zbl 0802.20024 |
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