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- W2169463973 abstract "The proof will be completed in Section 6. It is geometrical and the geometrical objects which will be used are graphs on which finite groups act (G-graphs). The following lines are intended to describe the structure of the proof with the main intermediate results. The translation into geometrical language is made possible by the realization theorem of Culler [S] which states that every finite subgroup of Out @ is realized by automorphisms of a graph (see Section 3). Now if G is a finite subgroup of Out Cp and r a graph realizing G then the centralizer of G can be identified with the group Out, n,(T, *) of equivariant conjugacy classes of automorphisms of rc,(f, *). We shall turn to fundamental groupoids and ‘prove with some effort (Sections 4 and 5) that if the G-graph r is suitably chosen (“reduced”) then the natural homomorphism Aut, Z7(r) + Out, z,(& *) is surjective. This way the problem reduces to the more tractable group Aut, n(r) of equivariant automorphisms of the free groupoid n(r). That this group is finitely generated will be proved in Section 6 by an extension of the Nielsen method in free groups. A peculiarity which occurs in the study of Aut, IZ(IJ is that we are forced to consider simultaneously a number of graphs related to r and all isomorphisms between fundamental groupoids of these graphs. In the classical situation of automorphisms of a free group (G trivial, r a wedge of loops) one proves that every automorphism of @ (i.e., of n(r)) can be expressed as a product of Nielsen automorphisms. What we have in the general situation is that (under certain circumstances described in Sec-" @default.
- W2169463973 created "2016-06-24" @default.
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- W2169463973 date "1989-07-01" @default.
- W2169463973 modified "2023-09-24" @default.
- W2169463973 title "Actions of finite groups on graphs and related automorphisms of free groups" @default.
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- W2169463973 doi "https://doi.org/10.1016/0021-8693(89)90154-3" @default.
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