Note on the Structure of $\text{P}\Sigma _n$

Abstract

The pure symmetric automorphism group, $\text{P}\Sigma _n$, consists of those automorphisms of the free group on $n$ generators that take each standard generator to a conjugate of itself. We give presentations for kernels of homomorphisms $\text{P}\Sigma _n\to \mathbb{Z}$ where each standard generator is sent to either 0 or 1, and provide explicit generators (as words in the standard generators) when those kernels are finitely generated. In addition, we provide recursive constructions of the defining graphs of the graph groups associated with $\text{P}\Sigma _n$.