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Boundaries of Croke–Kleiner‐admissible groups and equivariant cell‐like equivalence

Question 2.6 of Bestvina's Questions in Geometric Group Theory asks whether every pair of boundaries of a given CAT(0) group G is cell‐like equivalent [M. Bestvina, Questions in geometric group theory, http://www.math.utah.edu/~bestvina/eprints/questions‐updated.pdf]. The question was posed by Bestvina shortly after the discovery, by Croke and Kleiner [‘Spaces with nonpositive curvature and their ideal boundaries’, Topology 39 (2000) 549–556], of a CAT(0) group Γ that admits multiple boundaries. Previously, it had been observed by Bestvina and Geoghegan that all boundaries of a torsion‐free CAT(0) G would necessarily have the same shape. Since ‘cell‐like equivalence’ is weaker than topological equivalence, but in most circumstances, stronger (and more intuitive) than shape equivalence, this question is a natural one when working with the pathological types of spaces that occur as group boundaries. Furthermore, the definition of cell‐like equivalence allows for an obvious G ‐equivariant extension. In private conversations, Bestvina has indicated a preference for the G ‐equivariant formulation of Q2.6.