Geodesic
Complementing a~subgroup of a~hyperbolic group by a~free factor
Algebra i logika, Tome 52 (2013) no. 3, pp. 332-351.

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Let $G$ be a hyperbolic group that is not almost cyclic and $H$ be its quasiconvex subgroup of infinite index. We find necessary and sufficient conditions of there being for $H$ a free subgroup $F$ of rank 2 in $G$ such that $F$ and $H$ generate a free product $F*H\subseteq G$. It is proved that $F*H$ is quasiconvex and that there exists an algorithm for verifying the conditions of the criterium given $G$ and $H$.
Keywords: hyperbolic group, quasiconvex subgroup, free product. 
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F. A. Dudkin; K. S. Sviridov. Complementing a~subgroup of a~hyperbolic group by a~free factor. Algebra i logika, Tome 52 (2013) no. 3, pp. 332-351. http://geodesic.mathdoc.fr/item/AL_2013_52_3_a3/

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