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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:
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