On certain elements of free groups
This paper deals with certain elements of a free group \(F_m\), of rank \(m\), on which the way an endomorphism acts determines the endomorphism as a monomorphism or automorphism and the like. More precisely an element \(w\in F_m\) was defined by Shpilrain to be a test word if for any endomorphism \(\varphi\colon F_m\to F_m\) the equality \(\varphi(w)=w\) implies that \(\varphi\) is an automorphism. Several examples of test words are known and are exhibited in this paper. Two definitions are given here generalizing the notion of a test word. Namely: A non-trivial element \(w\in F_m\) is an \(M\)-test word for \(F_m\) if for every endomorphism \(\varphi\) and monomorphism \(\psi\) of \(F_m\) the equation \(\varphi(w)=\psi(w)\) implies that \(\varphi\) is also a monomorphism. A non-trivial word \(w(x_1,\dots,x_n)\) is a \(C\)-test word in \(n\) letters for \(F_m\) if for any two \(n\)-tuples \((A_1,\dots,A_n)\), \((B_1,\dots,B_n)\) of elements of \(F_m\) the equality \(w(A_1,\dots,A_n)=w(B_1,\dots,B_n)\neq 1\) implies the existence of an element \(S\in F_m\) such that \(B_i=SA_iS^{-1}\) for all \(i=1,2,\dots,n\). It follows that any \(C\)-test word in \(n\) letters is both test word and \(M\)-test word for \(F_m\). The purpose of this note is to construct \(C\)-test words in \(n\) letters. So the main result is the following: For an arbitrary \(n\geq 2\) there exists a non-trivial word \(w_n(x_1,\dots,x_n)\) which is a \(C\)-test word in \(n\) letters for any free group \(F_m\) of rank \(m\geq 2\). In addition \(w_n(x_1,\dots,x_n)\) is not a proper power. There are two corollaries of this theorem one of which states that there exist two elements \(u_1,u_2\in F_m\) such that any monomorphism \(\psi\) of \(F_m\) is uniquely determined by \(\psi(u_1)\), \(\psi(u_2)\). The element \(w_n(x_1,\dots,x_n)\) of the theorem is for \(n=2\) \[ \begin{multlined} w_2(x_1,x_2)=[x^8_1,x^8_2]^{100} x_1[x^8_1,x^8_2]^{200} x_1[x^8_1,x^8_2]^{300} x^{-1}_1[x^8_1,x^8_2]x^{-1}_1\\ [x^8_1,x^8_2]^{500} x_2[x^8_1,x^8_2]^{600} x_2[x^8_1,x^8_2]^{700} x^{-1}_2[x^8_1,x^8_2]^{800} x^{-1}_2\end{multlined} \] and by induction for \(n\geq 3\), \(w_n(x_1,\dots,x_n)=w_2(w_{n-1}(x_1,\dots,x_{n-1}),w_{n-1}(x_2,\dots,x_n))\). The greatest part of the paper contains the proof that the constructed element has the required property.
scientific article; zbMATH DE number 1218666