The generalised word problem in hyperbolic and relatively hyperbolic groups

We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is...

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Bibliographic Details
Published in:Journal of algebra 2018-12, Vol.516, p.149-171
Main Authors: Ciobanu, Laura, Holt, Derek, Rees, Sarah
Format: Article
Language:English
Subjects:
Context-free language
Generalised word problem
Real-time Turing machine
Relatively hyperbolic group
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Summary:We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller–Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2018.09.008