Notes on the structure of \(\text{P}\Sigma_n\). (Q1773052)

Let \(F_n\) be the free group with basis \(X=\{x_1,\dots,x_n\}\). The pure symmetric automorphism group of \(F_n\), denoted by \(\text{P}\Sigma_n\), is the subgroup of \(\Aut(F_n )\) consisting of those automorphisms of \(F_n\) which send each \(x_i\) to a conjugate of itself. \textit{J.~McCool}, [Can. J. Math. 38, 1525-1529 (1986; Zbl 0613.20024)], has given a finite presentation of \(\text{P}\Sigma_n\). In the present paper the authors first give presentations for kernels of homomorphisms \(\theta\) into an infinite cyclic group \(\langle t\rangle\), where each generator is mapped to \(t\) or \(1\). Since the image is cyclic, these kernels contain the commutator subgroup of \(\text{P}\Sigma_n\). Second they study the structure of the underlying graph of the graph group associated with \(\text{P}\Sigma_n\). Lastly, in the authors' expression, they provide explicit finite generating sets for those homomorphisms \(\theta\) for which \(\ker\theta\) is finitely generated.