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

TL;DR

This paper provides a concise presentation for the group of pure symmetric outer automorphisms of a free product splitting, using topological methods and a theorem by K. S. Brown to derive the result.

Contribution

It introduces a new presentation for these automorphisms by analyzing their action on a specific subcomplex of Outer Space, which is shown to be simply connected.

Findings

The subcomplex of Outer Space is simply connected.

A presentation for the automorphism group is derived using Brown's theorem.

The approach links topological properties with algebraic automorphism groups.

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.