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Complementing a subgroup of a hyperbolic group by a free factor. (English. Russian original) Zbl 1301.20033
Algebra Logic 52, No. 3, 222-235 (2013); translation from Algebra Logika 52, No. 3, 332-351 (2013).
Let $$G$$ be a group and $$H$$ a subgroup. $$H$$ is ‘complemented by a free factor of rank $$n$$ in $$G$$’ if there exists a free subgroup $$F_n$$ of rank $$n$$ in $$G$$ for which $$\langle F_n,H\rangle=F_n*H$$.
The authors show: Theorem 1. Let $$G$$ be a non-elementary word-hyperbolic group and $$H$$ a quasiconvex subgroup of infinite index. Assume that $$G$$ is defined by a finite set $$X$$ of generators and by a list of defining relations such that the corresponding hyperbolicity constant $$\delta$$ is known. Suppose that $$H$$ is defined by a finite set of generating words in $$X^{\pm 1}$$ such that the corresponding hyperbolicity constant $$\delta'$$ is known. Then the problem of complementability for $$H$$ by a free factor in $$G$$ is algorithmically decidable.

##### 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 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory
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