On endomorphisms of free groups that preserve primitivity
An element of a free group \(F_n\) of rank \(n\) is called primitive if it can be included in a free basis of \(F_n\). This paper deals with endomorphisms of free groups of finite rank which preserve primitivity. It is proved that in many cases these endomorphisms are automorphisms. The main results include the theorem that if \(\varphi\) is an endomorphism of the free group \(F_2\), of rank 2, which preserves primitivity, i.e. maps a primitive element onto a primitive element, then \(\varphi\) is an automorphism. If the image \(\varphi(F_n)\) of an endomorphism \(\varphi\) which preserves primitivity contains a primitive pair (i.e. two independent elements which can be included in a basis of \(F_n\)), then the above theorem holds for arbitrary finite ranks \(n\geq 2\). The main theorems are stated and proved in a more general situation. The proofs are carried out combinatorially using cancellation arguments and the Whitehead automorphisms.
endomorphisms of free groups of finite rank
scientific article; zbMATH DE number 1295479