A Presentation for the Group of Pure Symmetric Outer Automorphisms of a Given Splitting of a Free Product
Harry Iveson
TL;DR
This work constructs a concise finite presentation for the group of pure symmetric outer automorphisms $Out_{\mathfrak{S}}(G)$ of a free product $G=G_{1}*\dots*G_{n}$, with $n\ge3$, by embedding the problem into a subcomplex $\mathcal{C}_{n}$ of a relative Outer Space and applying Brown's theorem. The authors define $\mathcal{C}_{n}$ via $\,\mathfrak{S}$-labellings and collapses, analyse the action of $Out_{\mathfrak{S}}(G)$ with a strict fundamental domain $\mathcal{D}_{n}$, and compute vertex stabilisers using Guirardel–Levitt and Bass–Jiang approaches. They establish the simple connectivity of $\mathcal{C}_{n}$ (via the Space of Domains and peak reduction) and extract a presentation for $Out_{\mathfrak{S}}(G)$, with detailed cases $n\ge5$, $n=4$, and $n=3$, linking to McCool’s and Collins–Gilbert-type results. The framework also introduces a robust Space of Domains to study intersections of domains and to implement relative Whitehead automorphisms, paving the way for potential extensions to splittings with a free factor $F_{k}$. Overall, the paper provides a computationally tractable, geometry-driven method to present automorphism groups of free-product splittings and highlights connections to established presentations in special cases.
Abstract
We give a concise presentation for the group of pure symmetric outer automorphisms of a given splitting of a free product $G_{1}\ast\dots\ast G_{n}$. These are the (outer) automorphisms which preserve the conjugacy classes of the free factors $G_{i}$. This is achieved by considering the action of these automorphisms on a particular subcomplex of `Outer Space', which we show to be simply connected. We then apply a theorem of K. S. Brown to extract our presentation.