Comparing topologies on the Morse boundary and quasi-isometry invariance

We compare several topologies on the Morse boundary ${\mathrm{\partial }}_{M}Y$ of a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ 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.