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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.

20F65 Geometric group theory
57M07 Topological methods in group theory
20F28 Automorphism groups of groups
20E22 Extensions, wreath products, and other compositions of groups
Full Text: DOI
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