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The generalised word problem in hyperbolic and relatively hyperbolic groups

Lookup NU author(s): Professor Sarah Rees



This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).


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 rela-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.

Publication metadata

Author(s): Ciobanu L, Holt D, Rees s

Publication type: Article

Publication status: Published

Journal: Journal of Algebra

Year: 2018

Volume: 516

Pages: 149-171

Print publication date: 15/12/2018

Online publication date: 17/09/2018

Acceptance date: 23/09/2018

Date deposited: 06/11/2018

ISSN (print): 0021-8693

ISSN (electronic): 1090-266X

Publisher: Elsevier


DOI: 10.1016/j.jalgebra.2018.09.008


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FN PP00P2-144681/1