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

TL;DR

This paper proves that all boundaries of certain CAT(0) groups, specifically Croke-Kleiner-Admissible groups, are G-equivariantly cell-like equivalent, resolving a key question in geometric group theory about boundary equivalences.

Contribution

It establishes that for Croke-Kleiner-Admissible groups, all CAT(0) boundaries are G-equivariantly cell-like equivalent, confirming a conjecture for this class of groups.

Findings

All boundaries of Croke-Kleiner-Admissible groups are G-equivariantly cell-like equivalent.

The result applies to the original Croke-Kleiner group, providing a positive answer to a longstanding question.

The paper extends understanding of boundary equivalences in CAT(0) groups.

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. 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.