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On the outer automorphisms of hyperbolic groups. (Sur les automorphismes extérieurs des groupes hyperboliques.) (French) Zbl 0877.20014

The content of this paper is described very well in the author’s abstract: “We prove that an infinite nilpotent group of outer automorphisms in any word-hyperbolic group fixes projectively an action on an \(R\)-tree. In particular, we give short proofs of the theorem that any outer automorphism of a free group has a fixed point in the compactified Culler-Vogtmann outer space, and of Scott’s conjecture of the rank of the fixed points subgroup of a free automorphism.” We may also state here two corollaries of the main theorem: First that almost asymptotically certain the group of outer automorphisms of a finitely presented group (on two generators) is finite. Here, a property \(P\) of a group on \(p\) generators and \(q\) relations is almost asymptotically certain if the ratio of the number of (isomorphism classes of) groups on \(p\) generators and \(q\) relations of lengths \(n_1,\dots,n_q\) which satisfy \(P\), by the total number tends to 1 as \(n_1,\dots,n_q\) tend to infinity. Second the result of M. Bestvina and M. Handel about the Scott conjecture that the rank of the group of the fixed points, of an automorphism of the free group of rank \(n\), does not exceed \(n\).

MSC:

20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20E08 Groups acting on trees
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F28 Automorphism groups of groups
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References:

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