Quasi-Mobius Homeomorphisms of Morse boundaries

TL;DR

This paper explores the relationship between Morse boundaries of proper geodesic metric spaces and quasi-isometries, establishing conditions under which boundary homeomorphisms correspond to quasi-isometries.

Contribution

It characterizes when a homeomorphism of Morse boundaries arises from a quasi-isometry, specifically identifying quasi-mobius and 2-stable conditions as necessary and sufficient.

Findings

Homeomorphisms induced by quasi-isometries are exactly the quasi-mobius and 2-stable ones.

Provides a characterization of Morse boundary homeomorphisms corresponding to quasi-isometries.

Advances understanding of boundary behavior in hyperbolic-like spaces.

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.