Sets of primitive elements in a free group.
Let \(F_n\) be the free group on \(n\) generators and \(X=\{x_1,\dots,x_{n}\}\) a standard basis for \(F_n\). A set \(U=\{u_1,u_2,\dots,u_m\}\), \(m\leq n\), is called a primitive set in \(F_n\) if it can be extended to a basis of \(F_n\). In this paper the authors study primitive sets from three classical viewpoints. 1. A method of Whitehead -- A 3-manifold \(M\) which is homeomorphic to the connected sum of \(n\) copies \(S^1\times S^2\) has its fundamental group isomorphic to \(F_n\). One defines (see for details the authors [J. Group Theory 13, No. 4, 601-611 (2010; Zbl 1206.20025)]) a certain set \(\Sigma=\{S_1,S_2,\dots,S_n\}\) of \(n\) disjoint 2-spheres in \(M\) to be the standard sphere basis for \(M\) which is in some sense dual to the standard basis for \(F_n\). A collection \(\tau=\{T_1,T_2,\dots,T_m\}\) of disjoint 2-spheres embedded in \(M\) is said to be a sphere basis if \(m=n\), and there is a diffeomorphism of \(M\) which sends \(S_j\) to \(T_j\). The collection \(\tau=\{T_1,T_2,\dots,T_m\}\) is said to be a primitive sphere set if \(m\leq n\) and there is a diffeomorphism of \(M\) which maps \(T_j\) onto \(S_j\). This is equivalent to the existence of spheres \(\{T_{m+1},\dots,T_n\}\) such that \(\{T_1,T_2,\dots,T_m,T_{m+1},\dots,T_n\}\) is a sphere basis. Let \(M\), \(\Sigma\) and \(F_n\) as above. Theorem. Let \(U=\{u_1,u_2,\dots,u_m\}\), \(m\leq n\), be a set of distinct reduced words in \(F_n\). Then \(U\) is a primitive set if and only if there exists closed paths \(\{\alpha_1,\alpha_2,\dots,\alpha_m\}\) in \(M\) and 2-spheres \(\{T_1,T_2,\dots,T_m\}\) in \(M\) such that: (1) the word obtained by the intersection of \(\alpha_j\) with \(\Sigma\) is precisely \(u_j\); (2) \(a_i\cap T_i\) is a singleton; (3) \(a_i\cap T_j=\emptyset\) if \(i\neq j\). 2. A method of Nielsen -- Here the authors, using Nielsen transformations, prove that the preimage of a primitive set contains a primitive set. Theorem. Let \(G\) be a free group of rank \(m\) with basis \(Y=\{y_1,y_2,\dots,y_m\}\) and suppose that \(\Phi\) is a homomorphism from \(F_n\) to \(G\). If \(\{y_1,y_2,\dots,y_p\}\) is in the image of \(\Phi\) then there is a basis \(\{u_1,u_2,\dots,u_n\}\) for \(F_n\) and some \(p'\geq p\) such that: \(\Phi(u_i)=y_i\) for \(1\leq i\leq p\); \(\Phi\) is injective on the subgroup generated by \(\{u_1,u_2,\dots,u_{p'}\}\); and \(\Phi(u_i)=1\) for \(p'<i\leq n\). This theorem slightly generalizes Proposition 2.12 in [\textit{R. C. Lyndon} and \textit{P. E. Schupp}, Combinatorial group theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 89. Berlin-Heidelberg-New York: Springer-Verlag (1977; Zbl 0368.20023)]. 3. A method of Stallings -- In [Invent. Math. 71, 551-565 (1983; Zbl 0521.20013)] \textit{J. R. Stallings} introduced the use of labeled directed graphs to describe subgroups of groups acting on trees. Here the authors use this technique to prove the following Theorem. Let \(U=\{u_1,u_2,\dots,u_m\}\) be a set of \(m\) reduced words in \(F_n\) such that each \(x_i\) appears in at least one \(u_j\). Let \(\Gamma\) be a reduced graph representing the subgroup generated by \(U\). Then \(U\) is a primitive set if and only if there is a sequence of pinches and folds which includes exactly \(n-m\) pinches that transforms \(\Gamma\) into the elementary wedge \(W\). In [Ann. Math. (2) 37, 782-800 (1936; Zbl 0015.24804)] \textit{J. H. C. Whitehead} gave an algorithm which determined whether \(U=\{u_1,u_2,\dots,u_m\}\) is a primitive set in \(F_n\). The last of the above theorems gives a simpler algorithm.
scientific article; zbMATH DE number 6104539