The paper under review gives examples that quasi-isometries of \(\mathrm{CAT}(0)\) spaces do not, in general, induce homeomorphisms of the contracting boundary with subspace topology. The notion of a quasi-isometry invariant of a contracting boundary for a \(\mathrm{CAT}(0)\) space was introduced by R. Charney and H. Sultan [J. Topol. 8, No. 1, 93–117 (2015; Zbl 1367.20043)] and it enjoys many of the properties satisfied by boundaries of hyperbolic spaces.
The author’s example produces a quasi-isometry between \(\mathrm{CAT}(0)\) spaces but does not induces a continuous map between contracting boundaries. Because of this, the author proposes the following question:
“If \(\phi\) is a quasi-isometry between proper geodesic spaces \(X\) and \(Y\) that have a cocompact isometry groups and such that \(X\) and \(Y\) are \(\mathrm{CAT}(0)\), must \(\partial_C\phi : \partial_C^{G_p} X \rightarrow \partial_C^{G_p} Y\) be a homeomorphism?”