Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups. (29th August 2017)

• Record Type: Journal Article
• Authors: Cordes, Matthew; Durham, Matthew Gentry
• Is Part Of: International mathematics research notices. Volume 2019:Number 6(2019)
• Page Start: 1699
• Page End: 1724
• Publication Date: 2017-08-29
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx166
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: A Kleinian group $\Gamma < \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or, equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$ . Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.