The generalised word problem in hyperbolic and relatively hyperbolic groups

Laura Ciobanu, Derek Holt, Sarah Rees

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Abstract

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.
Original languageEnglish
Pages (from-to)149-171
Number of pages23
JournalJournal of Algebra
Volume516
Early online date17 Sep 2018
DOIs
Publication statusPublished - 15 Dec 2018

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