  We provide a new method of constructing non-quasiconvex subgroups of
hyperbolic groups by utilizing techniques inspired by Stallings' foldings. The
hyperbolic groups constructed are in the natural class of right-angled Coxeter
groups (RACGs for short) and can be chosen to be 2-dimensional. More
specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic)
RACG, our construction gives a corresponding non-quasiconvex subgroup of a
hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex
subgroups of hyperbolic RACGs including subgroups whose generators are as short
as possible (length two words), finitely generated free subgroups, non-finitely
presentable subgroups, and subgroups of fundamental groups of square complexes
of nonpositive sectional curvature.
