On strongly quasiconvex subgroups

Abstract

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 quasiisometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties 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 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 non-trivial strongly quasiconvex subgroups of infinite index (i.e. non-trivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

The local-to-global property for Morse quasi-geodesics

“…Stable subgroups are a strong generalization of quasiconvex subgroups that requires every pair of elements to be joined by a uniform Morse geodesic that stays uniformly close to the subgroup. Stable subgroups were introduced by Durham and Taylor to study the convex cocompact subgroups of the mapping class group [26], but have sense been studied in a variety of non-hyperbolic groups [1,34,45]. Theorem G is particularly interesting in the mapping class group, where it produces combination theorems for the convex cocompact subgroups by the work of Durham and Taylor (see Sect.…”

Quasi‐Mobius Homeomorphisms of Morse boundaries

“…The Morse boundary is designed to behave like boundaries of hyperbolic groups and, hopefully, to have similar applications to more general groups. Evidence of this may be found in several papers including . For an overview of what is currently known about Morse boundaries, see Cordes’ survey paper .…”

Random walks and quasi‐convexity in acylindrically hyperbolic groups

“…Generally speaking, relatively quasi-convex subgroups do not have to be stable in G. For example, each peripheral subgroup P i is relatively quasi-convex, but P i will be stable in G if and only if it is hyperbolic. In fact, if each peripheral subgroup is one-ended with linear divergence then a relatively quasi-convex subgroup of G is stable if and only if it has a finite intersection with each conjugate of each P i by [9, Theorem 5.4; 59, Theorem 4.13]; see [66] for more on stable subgroups of relatively hyperbolic groups.…”