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Equivariant outer space and automorphisms of free-by-finite groups. (English) Zbl 0805.20030

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)].

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
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