## The generalised word problem in hyperbolic and relatively hyperbolic groups

### Laura Ciobanu, Derek Holt, and Sarah Rees

#### Keywords

word problem, generalised word problem, hyperbolic group, relatively hyperbolic groups, real-time Turing machine context-free language, pushdown automaton

#### Status

Submitted for publication.

#### 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.
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