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

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.