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

TL;DR

This paper establishes conditions under which boundary components of cocompact convex sets in CAT(0)-spaces are contractible and explores the topological nature of limit sets of certain subgroups in negatively curved manifolds.

Contribution

It introduces new criteria for boundary contractibility in CAT(0)-spaces and analyzes the wildness of limit sets of quasi-convex subgroups in negatively curved manifolds.

Findings

Boundary components of cocompact convex sets can be contractible under certain conditions.

Limit sets of some subgroups may be 'wild' in the boundary, depending on geometric properties.

Coarse curvature bounds and barycenter techniques are effective tools in this analysis.

Abstract

We provide sufficient conditions as to when a boundary component of a cocompact convex set in a CAT(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.