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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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