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G-graphs and automorphisms of (virtually) free groups

Actions of finite groups on graphs and related automorphisms of free groups, J. of Algebra 124 (1989), 119-138.

A uniqueness decomposition theorem for finite actions on free groups, J. of Pure and Appl. Algebra 61 (1989), 29-48.

Finitely generated virtually free groups have finitely presented automorphism group, Proc. London Math. Soc. 64 (1992), 49-69.

Equivariant outer space and automorphisms of free-by-finite groups (with K. Vogtmann), Commentarii Math. Helvetici, 68 (1993), 216-262.

An equivariant Whitehead algorithm and conjugacy for roots of Dehn twists (with M. Lustig and K. Vogtmann), Proc. Edinburgh Math. Soc. (to appear).


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 [1], 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 [1] contains a direct proof of that isomorphism problem.

Another consequence of [1] is that the conjugacy problem for periodic elements and finite subgroups is solvable in both Aut(Fn) and Out(Fn). In [2], 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 $n\le 5$, and also a classification of the elements of prime order in Aut(Fn) for any n.

In [3], 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 [3] and [CV], K. Vogtmann and I show in [4] 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]: $vcd\, Out(G_1\ast\cdots\ast G_n) = n-2$, where Gi are non-trivial finite groups. (McCullough and Miller have an independent proof [MM].)

The paper [5] 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].


next up previous
Next: The groups IA(Fn) and GLn Up: SUMMARY OF RESULTS Previous: SUMMARY OF RESULTS