The generalised word problem for subgroups of hyperbolic groups

TL;DR

This paper investigates the computational complexity of the generalized word problem in hyperbolic groups, establishing conditions under which it is context-free or recognized by real-time Turing machines, and characterizing hyperbolic groups with certain subgroups.

Contribution

It proves the context-freeness of the generalized word problem for subgroups of virtually free groups and characterizes hyperbolic groups with specific quasiconvex subgroups.

Findings

Generalized word problem is context-free for subgroups of virtually free groups.

Hyperbolic groups with certain quasiconvex subgroups are characterized as virtually free.

The generalized word problem for torsion-free quasiconvex subgroups is recognized by a real-time Turing machine.

Abstract

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, 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, and that, for any hyperbolic group, the generalised word problem of a torsion-free quasiconvex subgroup is recognised by a real-time Turing machine.