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On a theorem of Scott and Swarup. (English) Zbl 0918.20028
Let \(G\) be a word hyperbolic group and let \(H\) be a subgroup of \(G\). The author says that \(H\) is distorted if \(H\) is not quasiconvex in \(G\). The author uses then the notion of a hyperbolic automorphism \(\phi\) of a word hyperbolic group \(H\), which is due to M. Bestvina and M. Feighn and which is contained in their paper [in J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)]. Such an automorphism gives rise to an exact sequence of hyperbolic groups \(1\to H\to G\to Z\to 1\). (Bestvina and Feighn show in fact that the automorphism \(\phi\) is hyperbolic if and only if \(G\) is word hyperbolic.)
In this paper, the author proves the following Theorem. Let \(1\to H\to G\to Z\to 1\) be an exact sequence of word hyperbolic groups induced by a hyperbolic automorphism \(\phi\) of the free group \(H\). Let \(H_1(\subset H)\) be a finitely generated distorted subgroup of \(G\). Then there exists \(N>0\) and a free factor \(K\) of \(H\) such that the conjugacy class of \(K\) is preserved by \(\phi^N\) and \(H_1\) contains a finite index subgroup of a conjugate of \(K\).
As a corollary, the authors obtain the following Theorem. Let \(1\to H\to G\to Z\to 1\) be an exact sequence of word hyperbolic groups induced by an aperiodic automorphism of the free group \(H\), and let \(H_1\) be a finitely generated subgroup of infinite index in \(H\). Then \(H_1\) is quasiconvex in \(G\).
The author’s results are an analog for free groups of a result of P. Scott and G. Swarup which states that if \(1\to H\to G\to Z\to 1\) is an exact sequence of word hyperbolic groups induced by a pseudo-Anosov automorphism of a closed surface with fundamental group \(H\), and if \(H_1\) is a finitely generated subgroup of infinite index in \(H\), then \(H_1\) is quasiconvex in \(G\). The techniques used involve the notions of laminations for free groups (developed mainly by Bestvina, Feighn and Handel), as well as the notions of Cannon-Thurston maps for hyperbolic group extensions, which were developed by the author in previous papers. The author shows in particular that for a distorted subgroup, the distorsion information is encoded in a certain set of “ending laminations”.

MSC:
20F65 Geometric group theory
57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
20E07 Subgroup theorems; subgroup growth
20E22 Extensions, wreath products, and other compositions of groups
20E36 Automorphisms of infinite groups
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