# Comparing topologies on the Morse boundary and quasi-isometry invariance

## TL;DR

This paper compares different topologies on the Morse boundary of CAT(0) cube complexes, revealing their differences and implications for quasi-isometry invariance, and introduces new descriptions and obstructions related to geodesic behavior.

## Contribution

It demonstrates that two topologies on the Morse boundary are generally not equal, provides a new cube complex perspective, and offers an obstruction to quasi-isometry invariance.

## Key findings

- Cashen and Mackay topologies are not equal in general
- A new description of one topology in cube complex language
- Obstruction to quasi-isometry invariance based on geodesic behavior

## 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.