Primitive elements in the free metabelian group of rank \(3\)
Let \(R_m=\mathbb{Z}[x_1^{\pm 1},\ldots,x_m^{\pm 1}]\) denote the Laurent polynomial ring in indeterminates \(x_1,\ldots,x_m\) with coefficients in \(\mathbb{Z}\). We let \(\text{SL}_n(R_m)\) denote the special linear group and \(E_n(R_m)\) denote the subgroup of \(\text{SL}_n(R_m)\) generated by the elementary matrices in \(\text{SL}_n(R_m)\). The well known existence theorem of \textit{S. Bachmuth} and \textit{H. Y. Mochizuki} [Am. J. Math. 104, 1181-1189 (1982; Zbl 0513.20038)] states that \(\text{SL}_2(R_m)\not=E_2(R_m)\) for \(m\geq 2\) (the case \(m=1\) is still open). This is an exceptional property because a famous result of \textit{A. A. Suslin} [Izv. Akad. Nauk SSSR, Ser. Mat. 41, 235-252 (1977; Zbl 0354.13009)] asserts that \(\text{SL}_n(R_m)=E_n(R_m)\) for \(n\geq 3\) and every \(m\). Let \(F_n\) be the free group and \(M_n\) be the free metabelian group of rank \(n\). It is well known from the results of Bachmuth and Mochizuki (and of the reviewer) that every automorphism of \(M_n\) is induced by some automorphism of \(F_n\) for \(n\not=3\). The case \(n=3\) is exceptional. \textit{S. Bachmuth} and \textit{H. Y. Mochizuki} [J. Algebra 55, 106-115 (1978; Zbl 0401.20033)] proved that \(\Aut M_3\) is infinitely generated. Moreover, the reviewer [\textit{V. A. Roman'kov}, Mat. Sb. 182, No. 7, 1074-1085 (1991; Zbl 0743.20035), English transl. in Math. USSR, Sb. 73, No. 2, 445-455 (1992)] proved that there exists a primitive element of \(M_3\) that is not the image of a primitive element of \(F_3\) under the natural map from \(F_3\) to \(M_3\). No specific elements of this type have appeared in the literature and the main purpose of the paper is to exhibit such one. Thus, the author gives examples of matrices in \(\text{SL}_2(R_2)\setminus E_2(R_2)\), and examples of specific primitive elements in \(M_3\) which are not induced by primitive elements in \(F_3\).
scientific article; zbMATH DE number 1384470