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Finite groups of outer automorphisms of a free group. (English) Zbl 0552.20024

Contributions to group theory, Contemp. Math. 33, 197-207 (1984).
[For the entire collection see Zbl 0539.00007.]
The author uses a geomeric approach to study arbitrary finite subgroups of the outer automorphism group of a free group. Let X be an arc- connected space and \(\pi_ 1(X)\) its fundamental group; then any homeomorphism of X induces a unique outer automorphism of \(\pi_ 1(X)\), defining in this way a homomorphism \(I_*: {\mathcal H}(X)\to Out(\pi_ 1(X))\), where \({\mathcal H}(X)\) is the group of homeomorphisms of X. If \(\pi\) is a group, \(\Gamma\) a subgroup of Out(\(\pi)\), then we say that \(\Gamma\) is realized by a group of homeomorphisms of X, if there exists a subgroup G of \({\mathcal H}(X)\) and an isomorphism \(\phi\) : \(\pi\) \({}_ 1(X)\to \pi\), such that \(\phi_*\circ I_*\) restricts to an isomorphism of G to \(\Gamma\), where \(\phi_*: Out(\pi_ 1(X))\to Out(\pi)\) is induced by \(\phi\). One of the main results of the paper is that: If F is a free group, then any finite subgroup of Out(F) is realized by a group of automorphism of a graph. If F is finitely generated, then the graph can be taken to be compact. The author uses his methods to study the problem of lifting groups of outer automorphisms to groups of automorphisms.
Reviewer: S.Andreadakis

MSC:

20F28 Automorphism groups of groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20E05 Free nonabelian groups
57S17 Finite transformation groups

Citations:

Zbl 0539.00007