## Hyperbolic groups and free constructions.(English)Zbl 0902.20018

The authors make the following definitions: A subgroup $$U$$ of a group $$G$$ is said to be conjugate separated if the set $$\{u\in U\mid u^x\in U\}$$ is finite for all $$x\in G\setminus U$$. Suppose now that $$U$$ and $$V$$ are subgroups of $$G$$, let $$\psi\colon U\to V$$ be an isomorphism, that either $$U$$ or $$V$$ is conjugate separated and that the set $$\{U\cap g^{-1}Vg\}$$ is finite for all $$g\in G$$. Then, the HNN-extension $$\langle G,t\mid t^{-1}ut=u^\psi,\;u\in U\rangle$$ is said to be separated.
The authors then prove the following Theorem 1. If $$G$$ is a hyperbolic group and $$H=\langle G,t\mid U^t=V\rangle$$ is a separated HNN-extension such that the subgroups $$U$$ and $$V$$ are quasiisometrically embedded in $$G$$, then $$H$$ is hyperbolic. – Theorem 2. Let $$G_1$$ and $$G_2$$ be hyperbolic groups, with $$U\leq G_1$$ and $$V\leq G_2$$ quasiisometrically embedded, and $$U$$ conjugate separated in $$G_1$$. Then the group $$G_1*_{U=V}G_2$$ is hyperbolic.
They obtain the following corollaries: Corollary 1. If $$G$$ is a hyperbolic group with $$A$$ and $$B$$ isomorphic virtually cyclic subgroups, then the HNN-extension $$H=\langle G,t\mid A^t=B\rangle$$ is hyperbolic if and only if it is separated. – Corollary 2. Let $$G_1$$ and $$G_2$$ be hyperbolic groups, with $$A\leq G_1$$, $$B\leq G_2$$ virtually cyclic. Then the group $$G_1*_{A=B}G_2$$ is hyperbolic if and only if either $$A$$ is conjugate separated in $$G_1$$ or $$B$$ is conjugate separated in $$G_2$$. Corollary 2 has been proved by M. Bestvina and M. Feighn [in J. Differ. Geom. 35, No. 1, 85-102 (1992; Zbl 0724.57029)]. The authors note that Corollary 1 contradicts an assertion in this same paper.
The authors study then quasiconvexity, and they prove the following Theorem 3. Let $$H=\langle G,t\mid U^t=V\rangle$$ be hyperbolic with $$U$$ quasiconvex in $$H$$. Then, $$G$$ is quasiconvex in $$H$$ and hence hyperbolic. – Theorem 4. Let $$H$$ be a separated HNN-extension, $$H=\langle G,t\mid U^t=V\rangle$$, with $$G$$ hyperbolic and $$U$$ and $$V$$ quasiconvex in $$G$$. Then $$G$$ is quasiconvex in $$H$$.
Finally, the authors describe the $$Q$$-completion $$G^Q$$ of a torsion-free hyperbolic group $$G$$ (where $$Q$$ is the field of rationals) as a union of an effective chain of hyperbolic subgroups, and they give a solution for the word problem and the conjugacy problem in $$G^Q$$.

### MSC:

 20F65 Geometric group theory 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 57M07 Topological methods in group theory

### Citations:

Zbl 0724.57029; Zbl 0746.57021
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### References:

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