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.
Categories: math.GR