We develop a theory of
strongly quasiconvex subgroups of an arbitrary finitely
generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic
groups and is preserved under quasi-isometry. We show that strongly quasiconvex
subgroups are also more reflective of the ambient group’s geometry than the stable
subgroups defined by Durham and Taylor, while still having many properties
analogous to those of quasiconvex subgroups of hyperbolic groups. We characterize
strongly quasiconvex subgroups in terms of the lower relative divergence of ambient
groups with respect to them.
We also study strong quasiconvexity and stability in relatively hyperbolic groups,
right-angled Coxeter groups, and right-angled Artin groups. We give complete
descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and
we characterize strongly quasiconvex special subgroups and stable special subgroups
of two-dimensional right-angled Coxeter groups. In the case of right-angled Artin
groups, we prove that the two notions of strong quasiconvexity and stability are
equivalent when the right-angled Artin group is one-ended and the subgroups have
infinite index. We also characterize nontrivial strongly quasiconvex subgroups of
infinite index (ie nontrivial stable subgroups) in right-angled Artin groups by
quadratic lower relative divergence, expanding the work of Koberda, Mangahas, and
Taylor on characterizing purely loxodromic subgroups of right-angled Artin
groups.
