Some Results on Free Groups in Combinatorial Group Theory
Let Fm be a free group of a finite rank m ≥ 2. We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign