The generalised word problem in hyperbolic and relatively hyperbolic groups
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.
Cite
@article{arxiv.1511.00548,
title = {The generalised word problem in hyperbolic and relatively hyperbolic groups},
author = {Laura Ciobanu and Derek Holt and Sarah Rees},
journal= {arXiv preprint arXiv:1511.00548},
year = {2016}
}
Comments
This paper includes all the material from the preprint The generalised word problem for subgroups of hyperbolic groups (Derek F Holt and Sarah Rees) previously deposited on the arXiv as arXiv:1505.02397, and citations of that article should be replaced by citations of this current one