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The generalised word problem in hyperbolic and relatively hyperbolic groups
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Ciobanu, Laura, Holt, Derek F. and Rees, Sarah (2018) The generalised word problem in hyperbolic and relatively hyperbolic groups. Journal of Algebra, 516 . pp. 149-171. doi:10.1016/j.jalgebra.2018.09.008 ISSN 0021-8693.
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Official URL: https://doi.org/10.1016/j.jalgebra.2018.09.008
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.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Word problems (Mathematics), Hyperbolic groups, Turing machines | ||||||||
| Journal or Publication Title: | Journal of Algebra | ||||||||
| Publisher: | Academic Press Inc Elsevier Science | ||||||||
| ISSN: | 0021-8693 | ||||||||
| Official Date: | 15 December 2018 | ||||||||
| Dates: |
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| Volume: | 516 | ||||||||
| Page Range: | pp. 149-171 | ||||||||
| DOI: | 10.1016/j.jalgebra.2018.09.008 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Restricted or Subscription Access | ||||||||
| Date of first compliant deposit: | 12 September 2018 | ||||||||
| Date of first compliant Open Access: | 17 September 2019 | ||||||||
| RIOXX Funder/Project Grant: |
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