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.

The local-to-global property for Morse quasi-geodesics

“…1 The study of Morse quasi-geodesics arose from trying to understand the "hyperbolic directions" in a non-hyperbolic space [20] and has since received immense interest in the literature (see [1,2,23,34,40] for a sampling). Numerous results from hyperbolic spaces have fruitful generalizations to spaces containing infinite Morse quasi-geodesics, particularly with respect to the study of stable subgroups [5,7,16] and the quasi-isometric classification of spaces [15,17]. Our new contribution to the study of Morse quasi-geodesics is the introduction of a local-to-global property for Morse quasi-geodesics.…”

Morse boundaries of graphs of groups with finite edge groups

“…Morse boundaries inherit many properties of the Gromov boundary (see e.g. [CH17], [CCM19], [Mur19], [Liu21], [Zal18]) and [CCS20] gives a topological description of certain Morse boundaries. A summary of properties of the Morse boundary can be found in [Cor19].…”

Cross Ratios on Boundaries of Symmetric Spaces and Euclidean Buildings

“…There is already some work done in this context. In [CM17], see also [CCM18], a coarse cross ratio for arbitrary CAT(0) spaces on some subset of the boundary has been constructed. For CAT(0) cube complexes there is a cross ratio on the Roller boundary constructed in [BFIM], using essentially the combinatorial structure of the space.…”