The generalised word problem in hyperbolic and relatively hyperbolic groups

TL;DR

This paper investigates the computational complexity of the generalized word problem in hyperbolic and relatively hyperbolic groups, showing it is solvable by real-time Turing machines or context-free under certain conditions.

Contribution

It extends known results by proving the generalized word problem is real-time solvable in relatively hyperbolic groups and characterizes hyperbolic groups with context-free problems.

Findings

Generalized word problem in relatively hyperbolic groups is real-time solvable.

In hyperbolic groups, the problem is real-time under specific subgroup conditions.

Hyperbolic groups with certain subgroups have a context-free generalized word problem.

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.