Symmetric Automorphisms of Free Products

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.