@MISC{Shpilrain96generalizedprimitive, author = {Vladimir Shpilrain}, title = {GENERALIZED PRIMITIVE ELEMENTS OF A FREE}, year = {1996} }

Share

OpenURL

Abstract

Abstract. We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts “like an automorphism ” on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological properties of the corresponding one-relator group. These elements (“generalized primitive elements”), interesting in their own right, can also be used for distinguishing automorphisms among arbitrary endomorphisms. Let F = Fn be the free group of a finite rank n ≥ 2 with a set X = {xi},1 ≤ i ≤ n, of free generators. An element g ∈ F is called primitive if it is a member of some free basis of F. Or, equivalently: there is an automorphism φ ∈ AutF that takes g to x1.