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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 $$G$$.

##### 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
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##### References:
  Sur les Groupes Hyperboliques D’aprés Mikhael Gromov, Progress Math., 83, Birkhaüser, Boston, MA (1990).  O. V. Bogopolskii and V. N. Gerasimov, ”Finite subgroups of hyperbolic groups,” Algebra Logika, 34, No. 6, 619–622 (1995).  N. Brady, ”Finite subgroups of hyperbolic groups,” Int. J. Alg. Comput., 10, No. 4, 399–405 (2000). · Zbl 1010.20030  M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundl. Math. Wiss., 319, Springer, Berlin (1999). · Zbl 0988.53001  R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin (1977).  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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