Homotopy equivalent boundaries of cube complexes

Abstract

A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary \(\partial _TX\) (which depends on the CAT(0) metric), the Roller boundary \({\partial _R}X\) (which depends only on the combinatorial structure), and the simplicial boundary \(\partial _\triangle X\) (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of \({\partial _R}X\) to define a simplicial Roller boundary \({\mathfrak {R}}_\triangle X\). Then, we show that \(\partial _TX\), \(\partial _\triangle X\), and \({\mathfrak {R}}_\triangle X\) are all homotopy equivalent, \(\text {Aut}(X)\)-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.