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A hyperbolic-by-hyperbolic hyperbolic group. (English) Zbl 0895.20028
Given a short exact sequence of finitely generated groups $$1\to K\to G\to H\to 1$$, the author has shown [in J. Pure Appl. Algebra 110, No. 3, 305-314 (1996; Zbl 0851.20037)] that if $$K$$ and $$G$$ are word hyperbolic and if $$K$$ is nonelementary (that is, not virtually cyclic), then $$H$$ is word hyperbolic.
The interesting original example of this situation is due to Thurston, and it involves the construction of the mapping torus of a pseudo-Anosov surface homeomorphism. Here, $$K$$ is the fundamental group of the surface, $$G$$ is the fundamental group of the resulting three-manifold (which is a hyperbolic manifold) and $$H$$ is the group $$Z$$ of integers. Other examples (which are generalizations of Thurston’s example) have been provided by Bestvina and Feighn, but in these examples also, $$H$$ is the group of integers.
In this paper, the author gives the first examples of such an exact sequence where each of $$K$$, $$G$$, $$H$$ is nonelementary. The construction involves also pseudo-Anosov mapping classes, and the main theorem which the author proves is the following:
Theorem: Let $$S$$ be a closed hyperbolic surface and let $$(S,p)$$ be the surface $$S$$ with one puncture $$p$$. Let $$\Phi_1,\dots,\Phi_n$$ be an independent set of pseudo-Anosov mapping classes of $$S$$ (that is, the fixed-point sets in the space of projective measured foliations are pairwise disjoint). Let $$i_1,\dots,i_n$$ be positive integers, and let $$H$$ be the subgroup of the mapping class group of $$S$$ which is generated by $$\Phi_1^{i_1},\dots,\Phi_n^{i_n}$$. Finally, let $$G$$ be the subgroup of the mapping class group of $$(S,p)$$ defined as the preimage of $$H$$ by the natural map. There is a short exact sequence $$1\to K\to G\to H\to 1$$ with $$K=\pi_1(S,p)$$. Then, if $$i_1,\dots,i_n$$ are sufficiently large, the group $$H$$ is free on the given generators and $$G$$ is word hyperbolic.
The paper contains some related open questions, for example whether the theorem above has a generalization when $$K$$ is replaced by an arbitrary word hyperbolic group.

##### MSC:
 20F65 Geometric group theory 57M07 Topological methods in group theory 20F28 Automorphism groups of groups 20E22 Extensions, wreath products, and other compositions of groups
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##### References:
  Andrew J. Casson and Steven A. Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Mathematical Society Student Texts, vol. 9, Cambridge University Press, Cambridge, 1988. · Zbl 0649.57008  M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85 – 101. · Zbl 0724.57029  M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geometric and Functional Analysis (1997), to appear. · Zbl 0884.57002  Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. · Zbl 0297.57001  Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107 – 1120. · Zbl 0551.57004 · doi:10.1215/S0012-7094-83-05046-9 · doi.org  J. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces , Oxford Univ. Press, 1991. CMP 92:02 · Zbl 0764.57002  J. Cannon and W. P. Thurston, Group invariant Peano curves, preprint. · Zbl 1136.57009  Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. · Zbl 0731.57001  H. Masur, 1994, private correspondence.  John McCarthy, A ”Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583 – 612. · Zbl 0579.57006  J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1 – 7. · Zbl 0162.25401  M. Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology (1997), to appear. · Zbl 0907.20038  Lee Mosher, Hyperbolic extensions of groups, J. Pure Appl. Algebra 110 (1996), no. 3, 305 – 314. · Zbl 0851.20037 · doi:10.1016/0022-4049(95)00081-X · doi.org  Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). · Zbl 0855.57003  E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Annals of Math. (1997), to appear. · Zbl 0910.57002  J. Stillwell, The Dehn-Nielsen theorem, Papers on group theory and topology, by M. Dehn (J. Stillwell transl.), Springer, 1987.
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