A hyperbolic-by-hyperbolic hyperbolic group.

*(English)*Zbl 0895.20028Given 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.

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.

Reviewer: A.Papadopoulos (Strasbourg)

##### 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 |

##### Keywords:

hyperbolic groups; pseudo-Anosov mapping classes; pseudo-Anosov homeomorphisms; finitely generated groups; word hyperbolic groups; fundamental groups; mapping class groups
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\textit{L. Mosher}, Proc. Am. Math. Soc. 125, No. 12, 3447--3455 (1997; Zbl 0895.20028)

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##### References:

[1] | 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 |

[2] | M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85 – 101. · Zbl 0724.57029 |

[3] | 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 |

[4] | 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 |

[5] | 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 |

[6] | 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 |

[7] | J. Cannon and W. P. Thurston, Group invariant Peano curves, preprint. · Zbl 1136.57009 |

[8] | 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 |

[9] | H. Masur, 1994, private correspondence. |

[10] | 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 |

[11] | J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1 – 7. · Zbl 0162.25401 |

[12] | M. Mitra, Cannon-Thurston maps for hyperbolic group extensions, Topology (1997), to appear. · Zbl 0907.20038 |

[13] | 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 |

[14] | 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 |

[15] | 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 |

[16] | J. Stillwell, The Dehn-Nielsen theorem, Papers on group theory and topology, by M. Dehn (J. Stillwell transl.), Springer, 1987. |

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