On the structure of two-generated hyperbolic groups.

*(English)*Zbl 0931.20036The authors prove the following Theorem A. Let \(G\) be a one-ended torsion free two-generated word-hyperbolic group and let \(G=\pi_1(A,T)\) be an essential cyclic splitting of \(G\), where \(A\) is a finite graph of groups and \(T\) a maximal tree in \(A\). Then, one of the following holds: 1. The graph \(A\) consists of a single vertex with no edges. 2. The graph \(A\) consists of a single vertex \(v\) and a single edge \(e\) with \(o(e)=t(e)=v\). In this case, the vertex group \(A_v\) is also one-ended two-generated and word-hyperbolic.

Theorem B. Let \(G\) be a one-ended two-generated torsion free word-hyperbolic. Then the group of outer automorphisms \(\text{Out}(G)\) contains a cyclic subgroup of finite index.

Theorem C. Two-generated torsion-free word-hyperbolic groups are strongly accessible.

We note that strong accessibility was introduced by B. Bowditch in his study of the boundary of word-hyperbolic groups. Theorem C means that two-generated torsion-free word-hyperbolic groups can be constructed from groups with no nontrivial cyclic splittings by applying finitely many times free products with amalgamation and HNN-extensions over cyclic groups.

The techniques in the proof are based on the JSJ decomposition theory, which has been intoduced by Z. Sela for torsion free hyperbolic groups and then extended by Z. Sela and E. Rips to finitely presented groups. For a freely indecomposable word-hyperbolic group \(G\), the JSJ decomposition is a presentation of \(G\) as the fundamental group of a graph of groups where the edge groups are infinite cyclic. This splitting gives in general information about the outer automorphism group of \(G\).

Theorem B. Let \(G\) be a one-ended two-generated torsion free word-hyperbolic. Then the group of outer automorphisms \(\text{Out}(G)\) contains a cyclic subgroup of finite index.

Theorem C. Two-generated torsion-free word-hyperbolic groups are strongly accessible.

We note that strong accessibility was introduced by B. Bowditch in his study of the boundary of word-hyperbolic groups. Theorem C means that two-generated torsion-free word-hyperbolic groups can be constructed from groups with no nontrivial cyclic splittings by applying finitely many times free products with amalgamation and HNN-extensions over cyclic groups.

The techniques in the proof are based on the JSJ decomposition theory, which has been intoduced by Z. Sela for torsion free hyperbolic groups and then extended by Z. Sela and E. Rips to finitely presented groups. For a freely indecomposable word-hyperbolic group \(G\), the JSJ decomposition is a presentation of \(G\) as the fundamental group of a graph of groups where the edge groups are infinite cyclic. This splitting gives in general information about the outer automorphism group of \(G\).

Reviewer: A.Papadopoulos (Strasbourg)

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20F05 | Generators, relations, and presentations of groups |

20F28 | Automorphism groups of groups |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |