Boundaries of Croke-Kleiner-admissible groups and equivariant cell-like equivalence (Q2921100)

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 [\textit{M. Bestvina}, ``Questions in geometric group theory'', \url{http://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf}]. The question was posed by Bestvina shortly after the discovery, by \textit{C. B. Croke} and \textit{B. Kleiner} [Topology 39, No. 3, 549--556 (2000; Zbl 0959.53014)], of a CAT(0) group \(\Gamma\) 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.NEWLINENEWLINEIn this paper, we provide a positive answer to Bestvina's \(G\)-equivariant Cell-like Equivalence Question for the class of admissible groups studied by Croke and Kleiner in 2000 [loc. cit.]. Since that collection includes the original Croke-Kleiner group \(\Gamma\), our result provides a strong solution to Q2.6, for the group that originally motivated the question.