Equivariant outer space and automorphisms of free-by-finite groups
The authors study the automorphism group of a finitely generated free-by- finite group. By results of \textit{J. McCool} [Bull. Lond. Math. Soc. 20, No. 2, 131-135 (1988; Zbl 0641.20028)] these groups can be studied by considering centralizers of finite subgroups of the group \(\text{Out}(F_ n)\) of the outer automorphisms of a finitely generated free group \(F_ n\). Using results of McCool and independently \textit{S. Kalajdžievski} [J. Algebra 150, No. 2, 435-502 (1992; Zbl 0780.20015)] and \textit{S. Krstić} [Proc. Lond. Math. Soc., III. Ser. 64, No. 1, 49-69 (1992; Zbl 0773.20008)] proved that these groups are finitely presented. In the present paper the authors, for a finite subgroup \(G\) of \(\text{Out}(F_ n)\), construct a simplicial complex \(L_ G\) on which the centralizer \(C(G)\) acts with finite stabilizers and finite quotient. This complex is an equivariant deformation retract of the fixed point subcomplex of outer space \(X_ n\). They prove that the complex \(L_ G\) is contractible, they compute the dimension of \(L_ G\) and thus they give an upper bound on the virtual cohomological dimension (vcd) of \(C(G)\). This implies that \(C(G)\) has finitely generated homology in all dimensions. These homological finiteness properties translate directly into similar properties for automorphism groups of free-by- finite groups. In particular they prove that the vcd of the outer automorphism group of a free product of \(n\) finite groups is equal to \(n - 2\), proving thus in the affirmative a conjecture of \textit{D. J. Collins} [Arch. Math. 50, No. 5, 385-390 (1988; Zbl 0654.20036)].
scientific article; zbMATH DE number 484346