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

• Introduction
• Whitehead posets and symmetric Whitehead automorphisms
• The complexes \(K(G)\) and \(K_0(G)\)
• Lemmas of reductivity
• Contractibility of \(K_0(G)\)
• The vertex stabilizers and other subgroups of \(Aut(G)\)
• Applications to groups of automorphisms
• Finite groups of automorphisms
• Actions on \(\mathbb R\)-trees
• References