On certain C-test words for free groups
Abstract
Let Fm be a free group of a finite rank m > 1 and Xi, Yj be elements in Fm. A non-empty word w(x1,..., xn) is called a C-test word in n letters for Fm if, whenever w(X1,..., Xn)=w(Y1,..., Yn) not equal to 1, the two n-tuples (X1,..., Xn) and (Y1,..., Yn) are conjugate in Fm. In this paper we construct, for each n > 1, a C-test word vn(x1,..., xn) with the additional property that vn(X1,..., Xn)=1 if and only if the subgroup of Fm generated by X1,..., Xn is cyclic. Making use of such words vm(x1,..., xm) and vm+1(x1,..., xm+1), we provide a positive solution to the following problem raised by Shpilrain: There exist two elements u1, u2 in Fm such that every endomorphism of Fm with non-cyclic image is completely determined by its values on u1, u2.
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