A Presentation for the Group of Pure Symmetric Outer Automorphisms of a Given Splitting of a Free Product

Abstract

We give a concise presentation for the group of pure symmetric outer automorphisms of a given splitting of a free product $G_1\ast \cdots \ast G_n$. 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.

Reframed results to be more general (pure symmetric outer automorphisms of an arbitrary free product decomposition, as opposed to outer automorphisms of a Grushko decomposition). This involved some minor changes to language and notation used throughout the paper