On the group of conjugating automorphisms of a free group
Denote by \({\mathfrak F}_n\) (\(n\geq 2\)) the free group with free generators \(x_1,\dots,x_n\). An automorphism in the group \(\Aut({\mathfrak F}_n)\) that maps any generator \(x_i\) into a word of the type \(W^{-1}_ix_{p_i}W_i\) (for some word \(W_i=W_i(x_1,\dots,x_n)\) in \({\mathfrak F}_n\)) is called conjugating and the subgroup of all such automorphisms is denoted by \({\mathfrak C}_n\). Those among the conjugating automorphisms given by \(x_i\mapsto W^{-1}_ix_iW_i\), are called basis-conjugating automorphisms; denote their subgroup by \({\mathfrak M}_n\). Further, denote by \({\mathfrak S}_n\) the symmetric group on the set \(X=\{x_1,\dots,x_n\}\) and let \({\mathfrak B}_n\) be the braid group, i.e. the subgroup in \({\mathfrak C}_n\) of all those automorphisms that map the product \(x_1x_2\cdot\ldots\cdot x_n\) into itself. The group \({\mathfrak B}_n\) contains a subgroup called the pure braid group \({\mathfrak N}_n\) and generated by the automorphisms \(\varphi_{i,j}=\psi_{j-1}\psi_{j-2}\cdots\psi_{i+1}\psi^2_i\psi_{i+1}\cdots\psi_{j-2}\psi_{j-1}\) for all \(1\leq i<j\leq n\), with the automorphisms \(\psi_k\) (\(k=1,\dots,n-1\)) given by the rules \(x_k\mapsto x_{k+1}\), \(x_{k+1}\mapsto x^{-1}_{k+1}x_kx_{k+1}\) and satisfying the relations \(\psi_k\psi_{k+1}\psi_k=\psi_{k+1}\psi_k\psi_{k+1}\) (\(k=1,\dots,n-2\)) and \(\psi_k\psi_l=\psi_l\psi_k\) (\(k,l=1,\dots,n-1\); \(| k-l|\geq 2\)). The author proves that the group \({\mathfrak C}_n\) is the semidirect product of the invariant subgroups \({\mathfrak M}_n\) and \({\mathfrak S}_n\). There are given (Theorem 1) a description of the generators of \({\mathfrak C}_n\) via those for \({\mathfrak B}_n\) and \({\mathfrak S}_n\), also expressions for the generators of \({\mathfrak N}_n\) via those for \({\mathfrak M}_n\) (Lemma 3). It is proved that the centers of \({\mathfrak C}_n\) and \({\mathfrak M}_n\) are trivial. Criteria are given (Lemmas 4 and 5) for an automorphism to belong to the commutator subgroup \([{\mathfrak C}_n,{\mathfrak C}_n]\), or to the subgroup \([{\mathfrak M}_n,{\mathfrak M}_n]\). As usual, denote \([G,G]\) by \(G'\). Then, the author proves (Lemma 6) that \({\mathfrak C}_n'\cap{\mathfrak B}_n={\mathfrak B}_n'\), \({\mathfrak N}_n\cap{\mathfrak M}_n'={\mathfrak N}_n'\), \({\mathfrak M}_n'\cap{\mathfrak B}_n={\mathfrak N}_n'\) and \({\mathfrak M}_n'\cap{\mathfrak B}_n'={\mathfrak N}_n'\). It is proved (Theorem 4) that the kernel of the Burau representation of the braid group \({\mathfrak B}_n\) by matrices of order \(n\) with matrix elements from the ring of Laurent polynomials in one variable and with integral coefficients belongs to \({\mathfrak N}_n'\). Also, any element from the kernel of the Gassner representation of the pure braid group \({\mathfrak N}_n\) by matrices of order \(n\), with matrix elements from the ring of Laurent polynomials in \(n\) variables and with integral coefficients, when any strand is pulled out becomes an element of the kernel of the Gassner representation of the group \({\mathfrak N}_{n-1}\).
scientific article; zbMATH DE number 1158873