Quasi-Mobius Homeomorphisms of Morse boundaries

Abstract

The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for $X,Y$ proper, cocompact spaces, a homeomorphism between their Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.

arXiv admin note: substantial text overlap with arXiv:1707.07028

References in corpus (1)

• A rank-one CAT(0) group is determined by its Morse boundary

Cited by in corpus (2)

• An embedding of the Morse boundary in the Martin boundary
• Möbius structures, quasi-metrics, and completeness