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Quasiconvexity and amalgams. (English) Zbl 0912.20031
Let $$G$$ be a word hyperbolic group equipped with a finite symmetric set of generators $$S$$, with corresponding Cayley graph $$\Gamma$$ equipped with its simplicial metric. Let $$A$$ be a subgroup of $$G$$. Then $$A$$ is said to be quasiconvex in $$G$$ if there exists $$\varepsilon\geq 0$$ such that for all $$a\in A$$, for every geodesic path in $$\Gamma$$ joining the identity to $$a$$ and for all $$x$$ on this geodesic path, we can find $$a'\in A$$ such that $$d(x,a')\leq\varepsilon$$. It is well-known that this property of $$A$$ is independent of the choice of the generating set $$S$$.
The author proves the following Theorem 1. Suppose that the hyperbolic group $$G$$ is an amalgamated product, $$G=A_{-1}*_C A_1$$, with $$C$$ being a cyclic subgroup, and let $$H$$ be a finitely generated subgroup of $$G$$. Then, the following are equivalent: (a) $$H$$ is quasiconvex in $$G$$; (b) for all $$g\in G$$ and for all $$i=1,-1$$, the subgroup $$gHg^{-1}\cap A_i$$ is quasiconvex in $$A_i$$. (As the author notes, $$A_1$$ and $$A_{-1}$$ are quasiconvex subgroups in $$G$$.)
The author proves then several consequences of Theorem 1, concerning in particular the so-called exponential groups which have been studied by A. Myasnikov and V. Remeslennikov. Other important consequences concern the groups which have property $$(Q)$$. We recall that a hyperbolic group $$G$$ is said to have property $$(Q)$$ if any finitely generated subgroup of $$G$$ is quasiconvex in $$G$$. The author shows how to obtain new groups with property $$(Q)$$ from existing ones.
He proves the following result using Theorem 1: Theorem 2. Suppose that $$G=A_{-1}*_C A_1$$ is word hyperbolic, with $$C$$ virtually cyclic and $$A_{-1}$$ and $$A_1$$ having property $$(Q)$$. Then, $$G$$ has property $$(Q)$$. The techniques used involve Bass-Serre trees, graphs of groups and normal forms.

##### MSC:
 20F65 Geometric group theory 20E07 Subgroup theorems; subgroup growth 57M07 Topological methods in group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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