Comparing topologies on the Morse boundary and quasi-isometry invariance

DOI10.1007/S10711-020-00553-3arXiv1903.07048MaRDI QIDQ6315743

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Publication date: 17 March 2019

Abstract: We compare several topologies on the Morse boundary $partia{l}_{M}Y$ of a $mathrmCAT(0)$ 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.

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