##
**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 \(G^Q\).

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 \(G^Q\).

Reviewer: A.Papadopoulos (Strasbourg)

### 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 |

### Keywords:

finitely generated groups; hyperbolic groups; Gromov hyperbolicity; quasiconvex subgroups; decision problems; word problem; conjugacy problem; HNN-extensions; rational completions of groups; chains of groups; conjugate separated subgroups
PDF
BibTeX
XML
Cite

\textit{O. Kharlampovich} and \textit{A. Myasnikov}, Trans. Am. Math. Soc. 350, No. 2, 571--613 (1998; Zbl 0902.20018)

### References:

[1] | Gilbert Baumslag, On free \cal\?-groups, Comm. Pure Appl. Math. 18 (1965), 25 – 30. · Zbl 0136.01204 |

[2] | Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217 – 303. · Zbl 0178.34901 |

[3] | G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams, J. Pure Appl. Algebra 76 (1991), no. 3, 229 – 316. · Zbl 0749.20006 |

[4] | G. Baumslag, S. M. Gersten, M. Shapiro, and H. Short, Automatic groups and amalgams — a survey, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 23, Springer, New York, 1992, pp. 179 – 194. · Zbl 0749.20017 |

[5] | M. Bestvina and M. Feighn, A combination theorem for negatively curved groups, J. Differential Geom. 35 (1992), no. 1, 85 – 101. · Zbl 0724.57029 |

[6] | S. M. Gersten and H. B. Short, Rational subgroups of biautomatic groups, Ann. of Math. (2) 134 (1991), no. 1, 125 – 158. · Zbl 0744.20035 |

[7] | R. Gitik, On combination theorems for negatively curved groups, Internat. J. Algebra Comput. 6 (1996), 751-760. CMP 97:04 |

[8] | M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75 – 263. · Zbl 0634.20015 |

[9] | I. Kapovich, On a theorem of G. Baumslag, Proc. Special Session Combinatorial Group Theory and Related Topics (Brooklyn, NY, 1994), Amer. Math. Soc., Providence, RI (to appear). |

[10] | Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons, Inc.], New York-London-Sydney, 1966. · Zbl 0138.25604 |

[11] | K. V. Mikhajlovskii and A. Yu. Ol’shanskii, Some constructions relating to hyperbolic groups, 1994, Proc. Int. Conf. on Cohomological and Geometric Methods in Group Theory (to appear). |

[12] | A. G. Myasnikov and V. N. Remeslennikov, Exponential groups. II: Extension of centralizers and tensor completion of csa-groups, Internat. J. Algebra Comput. 6 (1996), 687-712. CMP 97:04 · Zbl 0866.20014 |

[13] | A. Yu. Ol\(^{\prime}\)shanskiĭ, Periodic quotient groups of hyperbolic groups, Mat. Sb. 182 (1991), no. 4, 543 – 567 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 519 – 541. |

[14] | A. Yu. Ol\(^{\prime}\)shanskiĭ, On residualing homomorphisms and \?-subgroups of hyperbolic groups, Internat. J. Algebra Comput. 3 (1993), no. 4, 365 – 409. · Zbl 0830.20053 |

[15] | P. Papasoglu, Geometric methods in group theory, Ph.D. thesis, Columbia Univ., New York, 1993. |

[16] | Michael Shapiro, Automatic structure and graphs of groups, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 355 – 380. · Zbl 0798.20020 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.