Krstić, Sava; Vogtmann, Karen Equivariant outer space and automorphisms of free-by-finite groups. (English) Zbl 0805.20030 Comment. Math. Helv. 68, No. 2, 216-262 (1993). The authors study the automorphism group of a finitely generated free-by- finite group. By results of 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 S. Kalajdžievski [J. Algebra 150, No. 2, 435-502 (1992; Zbl 0780.20015)] and 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 D. J. Collins [Arch. Math. 50, No. 5, 385-390 (1988; Zbl 0654.20036)]. Reviewer: S.Andreadakis (Athens) Cited in 2 ReviewsCited in 19 Documents MSC: 20F28 Automorphism groups of groups 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20J05 Homological methods in group theory 20F05 Generators, relations, and presentations of groups 57M60 Group actions on manifolds and cell complexes in low dimensions Keywords:automorphism group; finitely generated free-by-finite group; centralizers of finite subgroups; outer automorphisms; finitely presented; simplicial complex; equivariant deformation retract; fixed point subcomplex; outer space; virtual cohomological dimension; finitely generated homology; homological finiteness; free product Citations:Zbl 0641.20028; Zbl 0780.20015; Zbl 0773.20008; Zbl 0654.20036 PDFBibTeX XMLCite \textit{S. Krstić} and \textit{K. Vogtmann}, Comment. Math. Helv. 68, No. 2, 216--262 (1993; Zbl 0805.20030) Full Text: DOI EuDML