# Symmetric Automorphisms of Free Products

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*Darryl McCullough; Andy Miller*

This memoir examines the automorphism group of a group \(G\) with a fixed free product decomposition \(G_1*\cdots *G_n\). An automorphism is called symmetric if it carries each factor \(G_i\) to a conjugate of a (possibly different) factor \(G_j\). The symmetric automorphisms form a group \(\Sigma Aut(G)\) which contains the inner automorphism group \(Inn(G)\). The quotient \(\Sigma Aut(G)/Inn(G)\) is the symmetric outer automorphism group \(\Sigma Out(G)\), a subgroup of \(Out(G)\). It coincides with \(Out(G)\) if the \(G_i\) are indecomposable and none of them is infinite cyclic. To study \(\Sigma Out(G)\), the authors construct an \((n-2)\)-dimensional simplicial complex \(K(G)\) which admits a simplicial action of \(Out(G)\). The stabilizer of one of its components is \(\Sigma Out(G)\), and the quotient is a finite complex. The authors prove that each component of \(K(G)\) is contractible and describe the vertex stabilizers as elementary constructs involving the groups \(G_i\) and \(Aut(G_i)\). From this information, two new structural descriptions of \(\Sigma Aut (G)\) are obtained. One identifies a normal subgroup in \(\Sigma Aut(G)\) of cohomological dimension \((n-1)\) and describes its quotient group, and the other presents \(\Sigma Aut (G)\) as an amalgam of some vertex stabilizers. Other applications concern torsion and homological finiteness properties of \(\Sigma Out (G)\) and give information about finite groups of symmetric automorphisms. The complex \(K(G)\) is shown to be equivariantly homotopy equivalent to a space of \(G\)-actions on \(\mathbb R\)-trees, although a simplicial topology rather than the Gromov topology must be used on the space of actions.

#### Table of Contents

# Table of Contents

## Symmetric Automorphisms of Free Products

- Table of Contents vii8 free
- Abstract viii9 free
- Introduction 110 free
- §1 Whitehead posets and symmetric Whitehead automorphisms 918 free
- §2 The complexes K(G) and K[sub(0)](G) 1827
- §3 Lemmas of reductivity 2635
- §4 Contractibility of K[sub(0)](G) 3443
- §5 The vertex stabilizers and other subgroups of Aut(G) 4655
- §6 Applications to groups of automorphisms 5867
- §7 Finite groups of automorphisms 7281
- §8 Actions on IR-trees 8190
- References 95104