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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:
 [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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