On the image of the Burau representation of the IA-automorphism group of a free group.
Let \(F_n\) be a free group of rank \(n\), \(n\geq 2\) and \(\Gamma_n(1)=F_n,\Gamma_n(2),\dots\) its lower central series. For each \(k\geq 0\), \(\mathcal A_n(k)\) denotes the kernel of the homomorphism \(\Aut F_n\to\Aut(F_n/\Gamma_n(k+1))\). The descending filtration \(\Aut F_n=\mathcal A_n(0)\geq\mathcal A_n(1)\geq\mathcal A_n(2)\geq\cdots\) is obtained. Although this filtration is called Johnson filtration, it was introduced and studied by \textit{S. Andreadakis} [in Proc. Lond. Math. Soc., III. Ser. 15, 239-268 (1965; Zbl 0135.04502)]. From then many authors were focused on the terms of this filtration and on the graded quotients \(gr^k(\mathcal A_n)=\mathcal A_n(k)/\mathcal A_n(k+1)\) and obtained interesting results with emphasis given to the first term \(IA_n=\mathcal A_n(1)\), the group of IA-automorphisms of \(F_n\), (see the references of the paper under consideration). If \(x_1,x_2,\dots,x_n\) are free generators of the free group \(F_n\) and \(\mathfrak b\colon F_n\to\langle s\rangle\) the homomorphism defined by \(x_i\to s\) on the infinite cyclic group \(\langle s\rangle\), then it is defined the Magnus representation \(\tau_B:=\tau_{\mathfrak b}\colon IA_n\to\text{GL}(n,\mathbb{Z}[s^{\pm 1}])\), which here is called the Bureau representation of \(IA_n\). In this paper the author using the Bureau representation contributes to the study of the terms \(\mathcal A_n(k)\) and \(\mathcal A_n'(k)\), the \(k\)-th term of the lower central series of \(IA_n=\mathcal A_n(1)\). Let \(\mathcal B_n(k)=\tau_B(\mathcal A_n(k))\), \(\mathcal B_n'(k)=\tau_B(\mathcal A_n'(k))\) and \(gr^k(\mathcal B_n')=\mathcal B_n'(k)/\mathcal B_n'(k+1)\). Here we quote some of the author's results. Theorem A. For any \(n\geq 2\), \(H_1(\tau_B(IA_n),\mathbb{Z})\cong\mathbb{Z}^{\oplus n(n-1)}\). Theorem B. For any \(n\geq 2\), \(gr^2(\mathcal B_n')\cong\mathbb{Z}^{\oplus(n^2-n-1)}\). Corollary. For \(n\geq 2\) and \(k\geq 2\), \(H_1(\mathcal A_n(k),\mathbb{Z})\supset\mathbb{Z}^{\oplus k(n^2-n-1)}\). Theorem C. For any \(k\geq 2\), \(gr^k(\mathcal B_2')\cong\mathbb{Z}\).
scientific article; zbMATH DE number 5669126