All CAT(0) Boundaries of Croke-Kleiner-Admissible Groups are Equivariantly CE Equivalent]

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. The question was posed by Bestvina shortly after the discovery, by Croke and Kleiner, of a CAT(0) group 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 a obvious G-equivariant extension. In 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 2002. Since that collection includes the original Croke-Kleiner group, our result provides a strong solution to Q2.6, for the group that originally motivated the question.