# Symmetric automorphisms of free products

### About this Title

**Darryl McCullough** and **Andy Miller**

Publication: Memoirs of the American Mathematical Society

Publication Year
1996: Volume 122, Number 582

ISBNs: 978-0-8218-0459-9 (print); 978-1-4704-0167-2 (online)

DOI: http://dx.doi.org/10.1090/memo/0582

MathSciNet review: 1329943

MSC: Primary 20F28; Secondary 20E06, 57M07

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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.

Readership

Graduate students and research mathematicians interested
in infinite groups, particularly in topological and homological methods
in group theory.

### Table of Contents

**Chapters**

- Introduction
- 1. Whitehead posets and symmetric Whitehead automorphisms
- 2. The complexes $K(G)$ and $K_0(G)$
- 3. Lemmas of reductivity
- 4. Contractibility of $K_0(G)$
- 5. The vertex stabilizers and other subgroups of $Aut(G)$
- 6. Applications to groups of automorphisms
- 7. Finite groups of automorphisms
- 8. Actions on $\mathbb {R}$-trees