Culler's Realization Theorem states that every finite subgroup of Out(Fn)is realized by a group of symmetries of a finite graph [Cu]. In , I give a uniqueness companion to Culler's Theorem; it says that the minimal realizing graph is unique up to ``Nielsen equivalence''. In other words, this is a classification of G-graphs up to equivariant homotopy equivalence. The paper also contains a stronger form of the same result, saying that centralizers of finite subgroups in Aut(Fn) and Out(Fn) are finitely generated. McCool derived in [M] that the automorphism groups of finitely generated virtually free groups are then also finitely generated. Karrass, Pietrowski and Solitar found that McCool's result is precisely the missing link in their incomplete proof that the isomorphism problem for the class of finitely generated virtually free groups is solvable. The paper  contains a direct proof of that isomorphism problem.
Another consequence of  is that the conjugacy problem for periodic elements and finite subgroups is solvable in both Aut(Fn) and Out(Fn). In , I prove that a free group, with a given action of a finite group on it, uniquely decomposes into a free product of invariant subgroups. This enables me to obtain a classification of periodic elements in Aut(Fn) for , and also a classification of the elements of prime order in Aut(Fn) for any n.
In , I prove that the centralizers of finite subgroups of Out(Fn) (and with them, the automorphism groups of virtually free groups) are finitely presented, as conjectured by Solitar and McCool in [M]. An independent proof was given by S. Kalajdzievski [K].
Combining the ideas of  and [CV], K. Vogtmann and I show in  that finite subgroups of Out(Fn)have contractible fixed subspaces in the Outer Space. (T. White has an independent proof [W].) We build a complex of Culler-Vogtmann type for the centralizer of a finite subgroup G of Out(Fn) and in favorable cases we can use it to compute the virtual cohomological dimension of the automorphism group of a given virtually free group. In particular, we prove D. J. Collins's Conjecture [Co]: , where Gi are non-trivial finite groups. (McCullough and Miller have an independent proof [MM].)
The paper  contains a further elaboration of the ideas of [1,3,4] and [CL], and its results constitute an important part of M. Lustig's solution of the conjugacy problem for Aut(Fn) [L].
[[CL]] M. M. Cohen and M. Lustig, The conjugacy problem for Dehn twist automorphisms of free groups, Commentarii Math. Helvetici (to appear).
[[Co]] D. J. Collins, Cohomological dimension and symmetric automorphisms of a free group, Comment. Math.Helv. 64 (1989), 44-61.
[[Cu]] M. Culler, Finite groups of outer automorphisms of free groups, Contemp. Math. 33 (1984), 197-207.
[[CV]] M. Culler and K. Vogtmann, Moduli of graphs and automorphisms of free groups, Inventiones Math. (1986), 91-119.
[[K]] S. Kalajdzievski, Automorphism group of a free group: centralizers and stabilizers, J. Algebra 150 (1992), 435-502.
[[L]] M. Lustig, Prime factorization and conjugacy problem in Out(Fn), preprint 1994.
[[M]] J. McCool, The automorphism groups of finite extensions of free groups, Bull. London Math. Soc. 20 (1988), 131-135.
[[MM]] D. McCullough and A. Miller, Symmetric automorphisms of free products, Memoirs of the Amer. Math. Soc., to appear.
[[W]] T. White, Free points of finite groups of free group automorphisms, Proc. Amer. Math. Soc. 118 (1993), 681-688.