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

TL;DR

This paper characterizes when a homeomorphism of Morse boundaries of cocompact CAT(0) spaces is induced by a quasi-isometry, linking boundary properties to geometric equivalences.

Contribution

It establishes a precise criterion involving quasi-mobius and 2-stability conditions for boundary homeomorphisms to correspond to quasi-isometries in cocompact CAT(0) spaces.

Findings

Homeomorphisms that are quasi-mobius and 2-stable induce quasi-isometries.

The Morse boundary captures hyperbolic-like behavior in CAT(0) spaces.

A boundary homeomorphism corresponds to a quasi-isometry if and only if it satisfies specific stability conditions.

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 cocompact CAT(0) spaces, a homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.