Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups.

Cordes, Matthew; Durham, Matthew Gentry

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.

KLEINIAN groups; SUBGROUP analysis (Experimental design); AUTOMORPHIC functions; HYPERBOLIC groups; CONVEX domains

IMRN: International Mathematics Research Notices, 2019, Vol 2019, Issue 6, p1699

1073-7928

Academic Journal

10.1093/imrn/rnx166