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A hyperbolic-by-hyperbolic hyperbolic group

Author: Lee Mosher
Journal: Proc. Amer. Math. Soc. 125 (1997), 3447-3455
MSC (1991): Primary 20F32; Secondary 57M07, 20F28
MathSciNet review: 1443845
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Abstract: Given a short exact sequence of finitely generated groups

\begin{displaymath}1 \to K \to G \to H \to 1 \end{displaymath}

it is known that if $K$ and $G$ are word hyperbolic, and if $K$ is nonelementary, then $H$ is word hyperbolic. In the original examples due to Thurston, as well as later examples due to Bestvina and Feighn, the group $H$ is elementary. We give a method for constructing examples where all three groups are nonelementary.

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  • [BC88] S. Bleiler and A. Casson, Automorphisms of surfaces after Nielsen and Thurston, LMS Student Texts, vol. 9, Cambridge University Press, 1988. MR 89k:57025
  • [BF92] M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Diff. Geom. 35 (1992), no. 1, 85-101. MR 93d:53053
  • [BFH96] M. Bestvina, M. Feighn, and M. Handel, Laminations, trees, and irreducible automorphisms of free groups, Geometric and Functional Analysis (1997), to appear.
  • [Bir74] J. Birman, Braids, links, and mapping class groups, Annals of Math. Studies, vol. 82, Princeton University Press, 1974. MR 51:11477
  • [BLM83] J. Birman, A. Lubotsky, and J. McCarthy, Abelian and solvable subgroups of the mapping class group, Duke Math. J. 50 (1983), no. 4, 1107-1120. MR 85k:20126
  • [Can91] J. Cannon, The theory of negatively curved spaces and groups, Ergodic theory, symbolic dynamics, and hyperbolic spaces (C. Series T. Bedford, M. Keane, ed.), Oxford Univ. Press, 1991. CMP 92:02
  • [CT] J. Cannon and W. P. Thurston, Group invariant Peano curves, preprint.
  • [FLP$^{+}$79] A. Fathi, F. Laudenbach, V. Poenaru, et al., Travaux de Thurston sur les surfaces, Astérisque, vol. 66-67, Société Mathématique de France, Paris, 1979. MR 82m:57003
  • [Mas94] H. Masur, 1994, private correspondence.
  • [McC85] J. McCarthy, A ``Tits-alternative'' for subgroups of surface mapping class groups, Trans. AMS 291 (1985), no. 2, 583-612. MR 87f:57011
  • [Mil68] J. Milnor, A note on curvature and fundamental group, J. Diff. Geom. 2 (1968), 1-7. MR 38:636
  • [Mit] M. Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology (1997), to appear.
  • [Mos96] L. Mosher, Hyperbolic extensions of groups, J. Pure and Appl. Alg. 110 (1996), no. 3, 305-314. MR 97c:20056
  • [Ota96] J.-P. Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3, Astérisque, no. 235, Société mathématique de france, 1996. MR 97e:57013
  • [RS95] E. Rips and Z. Sela, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Annals of Math. (1997), to appear.
  • [Sti87] J. Stillwell, The Dehn-Nielsen theorem, Papers on group theory and topology, by M. Dehn (J. Stillwell transl.), Springer, 1987.

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Additional Information

Lee Mosher
Affiliation: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102

Received by editor(s): May 4, 1996
Additional Notes: Partially supported by NSF grant # DMS-9204331
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1997 American Mathematical Society

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