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Question 2.6 of Bestvina's Questions in Geometric Group Theory asks whetherevery pair of boundaries of a given CAT(0) group G is cell-like equivalent. Thequestion was posed by Bestvina shortly after the discovery, by Croke andKleiner, of a CAT(0) group that admits multiple boundaries. Previously, it hadbeen observed by Bestvina and Geoghegan that all boundaries of a torsion freeCAT(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 onewhen working with the pathological types of spaces that occur as groupboundaries. Furthermore, the definition of cell-like equivalence allows for aobvious G-equivariant extension.  In this paper we provide a positive answer to Bestvina's G-equivariantCell-like Equivalence Question for the class of admissible groups studied byCroke and Kleiner in 2002. Since that collection includes the originalCroke-Kleiner group, our result provides a strong solution to Q2.6, for thegroup that originally motivated the question.

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