Boundaries of Croke–Kleiner-admissible groups and equivariant cell-like equivalence. (5th May 2014) - Guilbault, Craig R.; Mooney, Christopher P.
Abstract : 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. 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 2000. Since that collection includes the original Croke–Kleiner group Γ, our result provides a strong solution to Q2.6, for the group that originally motivated the question.