Contractibility of boundaries of cocompact convex sets and embeddings of limit sets

Abstract

We provide sufficient conditions as to when a boundary component of a cocompact convex set in a CAT(0){\rm{CAT}}\left(0)-space is contractible. We then use this to study when the limit set of a quasi-convex, codimension one subgroup of a negatively curved manifold group is “wild” in the boundary. The proof is based on a notion of coarse upper curvature bounds in terms of barycenters and the careful study of interpolation in geodesic metric spaces.

Keywords

• cat(0) spaces
• cartan-hadamard manifold
• barycenters
• contractibility
• quasi-convex codimension one subgroups
• 53c23
• 52a20 (primary)
• 51f30
• 20f65
• 20f67 (secondary)