Pulling back stability with applications to Out($F_n$) and relatively hyperbolic groups

Abstract

We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.

arXiv comment21 pages, 1 figure. Update includes some corrections and revisions. To appear in JLMS

References in corpus (9)

• Relative hyperbolicity and relative quasiconvexity for countable groups
• Convex cocompact subgroups of mapping class groups
• Contracting Boundaries of CAT(0) Spaces
• Word hyperbolic extensions of surface groups
• Convex cocompactness and stability in mapping class groups
• Hyperbolic extensions of free groups
• Stability and the Morse boundary
• Strongly Contracting Geodesics in Outer Space
• Negative curvature in graphical small cancellation groups

Cited by in corpus (8)

• On strongly quasiconvex subgroups
• A Metrizable Topology on the Contracting Boundary of a Group
• Stable Subgroups of the Genus Two Handlebody Group
• Stable subgroups and Morse subgroups in mapping class groups
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• Contracting Boundary of a Cusped Space
• The geometry of subgroup embeddings and asymptotic cones
• Characterizations of Stability via Morse Limit Sets