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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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References:
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