Comparing topologies on the Morse boundary and quasi-isometry invariance

Abstract

We compare several topologies on the Morse boundary ${\partial }_{M}Y$ of a $\mathrm{CAT}(0)$ cube complex $Y$. In particular, we show that the two topologies introduced by Cashen and Mackay are not equal in general and provide a new description of one of them in the language of cube complexes. As a corollary, we obtain a new approach to tackle the question whether the visual topology induces a quasi-isometry-invariant topology on the Morse boundary. This leads to an obstruction to quasi-isometry-invariance in terms of the behaviour of geodesics under quasi-isometries.

27 pages, 4 figures, comments welcome

References in corpus (1)

• A Metrizable Topology on the Contracting Boundary of a Group

Cited by in corpus (2)

• Sublinearly Morse Geodesics in CAT(0) Spaces: Lower Divergence and Hyperplane Characterization
• An embedding of the Morse boundary in the Martin boundary