On the Johnson filtration of the basis-conjugating automorphism group of a free group.
Let \(\mathcal A_n(k)\) denote the kernel of the natural homomorphism \(\Aut(F_n)\to\Aut(F_n/\Gamma_n(k+1))\), then is defined the descending central filtration \(\Aut(F_n)=\mathcal A_n(0)\supset\mathcal A_n(1)\supset\mathcal A_n(2)\supset\cdots\) with \(\mathcal A_n(1)=IA_n\). This descending filtration of \(\Aut(F_n)\) is called the Johnson filtration of \(\Aut(F_n)\) and was originally introduced and studied [in Proc. Lond. Math. Soc., III. Ser. 15, 239-268 (1965; Zbl 0135.04502)] by \textit{S. Andreadakis}. Although since then many papers are devoted to studying this filtration and the graded quotients \(gr^k(\mathcal A_n)=\mathcal A_n(k)/\mathcal A_n(k+1)\), it was not yet determined whether each of the \(\mathcal A_n(k)\) is finitely generated or not for \(k\geq 2\). Neither is it known whether the Abelianization \(\mathcal A_n(k)^{ab}\) of \(\mathcal A_n(k)\) is finitely generated for \(k\geq 2\). In the present paper the author, continuing his work on the study of these filtrations, considers a special subgroup of \(\Aut(F_n)\) and studies the relevant Johnson filtration. More precisely, let \(\{x_1,x_2,\dots,x_n\}\) be a basis of the free group \(F_n\), an automorphism \(\sigma\) of \(F_n\) such that \(x_i^\sigma\) is conjugate to \(x_i\), \(1\leq i\leq n\), is called a basis-conjugating automorphism of \(F_n\). The set \(P\Sigma_n\) of all basis-conjugating automorphisms is a subgroup of \(\Aut(F_n)\) and \(P\Sigma_n\subseteq IA_n\). In [Can. J. Math. 38, 1525-1529 (1986; Zbl 0613.20024)] is given by \textit{J. McCool} a finite presentation of \(P\Sigma_n\). Let \(\mathcal P_n(k)=P\Sigma_n\cap\mathcal A_n(k)\) for each \(k\geq 1\). The groups \(\mathcal P_n(k)\) form the descending central Johnson filtration \(P\Sigma_n=\mathcal P_n(1)\supset\mathcal P_n(2)\supset\mathcal P_n(3)\cdots\). If \(P\Sigma_n=P\Sigma_n^{(1)}\supset P\Sigma_n^{(2)}\supset P\Sigma_n^{(3)}\supset\cdots\) is the lower central series of \(P\Sigma_n\), then \(P\Sigma_n^{(k)}\subset\mathcal P_n(k)\) for every \(k\geq 1\). It is proved that the Abelianization of \(P\Sigma_n\) is a free Abelian group of rank \(n(n-1)\) and that \(P\Sigma_n^{(2)}=\mathcal P_n(2)\) and \(P\Sigma_n^{(3)}=\mathcal P_n(3)\). Using the Reidemeister-Schreier method the author obtains an infinite presentation of \(P\Sigma_n^{(2)}=\mathcal P_n(2)\) and proves his main result. Theorem. For \(n\geq 3\) and \(k\geq 2\), \((P\Sigma_n^{(k)})^{ab}\) contains infinitely many linearly independent elements. This theorem has as corollary that for \(n\geq 3\) and \(k\geq 2\), \(\mathcal P_n(k)^{ab}\) contains infinitely many linearly independent elements. Therefore for \(n\geq 3\) and \(k\geq 2\), \(\mathcal P_n(k)\) is not finitely generated.
scientific article; zbMATH DE number 6141589
scientific article