Complementing a finite subgroup of a hyperbolic group by a free factor.

*(English. Russian original)*Zbl 1255.20041
Algebra Logic 49, No. 4, 354-377 (2010); translation from Algebra Logika 49, No. 4, 520-554 (2010).

From the introduction: Torsion in hyperbolic groups is heavily understudied. To our knowledge, the only significant result in this direction was obtained by O. V. Bogopol’skij and V. N. Gerasimov [in Algebra Logic 34, No. 6, 343-345 (1995); translation from Algebra Logika 34, No. 6, 619-622 (1995; Zbl 0901.20022)], stating that every finite subgroup of a \(\delta\)-hyperbolic group is conjugate to a subgroup contained in a ball of radius \(2\delta+1\) with center in unity. In [N. Brady, Int. J. Algebra Comput. 10, No. 4, 399-405 (2000; Zbl 1010.20030)], a similar statement was derived by using other methods.

In this paper we continue to study torsion in hyperbolic groups. Our main result is the following: Theorem. Let \(G\) be a hyperbolic group, which is not almost cyclic, and \(H\) be a finite subgroup of the group \(G\). For a group \(G\) to contain a free subgroup \(F\) of rank two such that \(\langle F,H\rangle=F*H\), it is necessary and sufficient that for every nonidentity element \(h\) of \(H\) there exist an element \(g(h)\) of infinite order in \(G\) such that \(\langle g(h)\rangle\cap C_G(h)=\{1\}\), where \(C_G(h)\) is the centralizer of \(h\) in \(G\).

In this paper we continue to study torsion in hyperbolic groups. Our main result is the following: Theorem. Let \(G\) be a hyperbolic group, which is not almost cyclic, and \(H\) be a finite subgroup of the group \(G\). For a group \(G\) to contain a free subgroup \(F\) of rank two such that \(\langle F,H\rangle=F*H\), it is necessary and sufficient that for every nonidentity element \(h\) of \(H\) there exist an element \(g(h)\) of infinite order in \(G\) such that \(\langle g(h)\rangle\cap C_G(h)=\{1\}\), where \(C_G(h)\) is the centralizer of \(h\) in \(G\).

##### MSC:

20F67 | Hyperbolic groups and nonpositively curved groups |

20E07 | Subgroup theorems; subgroup growth |

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

##### Keywords:

torsion in hyperbolic groups; free products; finite subgroups; free subgroups; elements of infinite order
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\textit{K. S. Sviridov}, Algebra Logic 49, No. 4, 354--377 (2010; Zbl 1255.20041); translation from Algebra Logika 49, No. 4, 520--554 (2010)

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##### References:

[1] | Sur les Groupes Hyperboliques D’aprés Mikhael Gromov, Progress Math., 83, Birkhaüser, Boston, MA (1990). |

[2] | O. V. Bogopolskii and V. N. Gerasimov, ”Finite subgroups of hyperbolic groups,” Algebra Logika, 34, No. 6, 619–622 (1995). |

[3] | N. Brady, ”Finite subgroups of hyperbolic groups,” Int. J. Alg. Comput., 10, No. 4, 399–405 (2000). · Zbl 1010.20030 |

[4] | M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundl. Math. Wiss., 319, Springer, Berlin (1999). · Zbl 0988.53001 |

[5] | R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin (1977). |

[6] | S. Billington, D. Epstein, and D. Holt, ”Geodesics in word hyperbolic groups,” http://www.maths.warwick.ac.uk/dbae/papers/geod.ps . |

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