A finite-dimensional CAT(0) cube complex
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Abstract X is equipped with several well-studied boundaries. These include theTits boundary (which depends on the CAT(0) metric), the$$\partial _TX$$ Roller boundary (which depends only on the combinatorial structure), and the$${\partial _R}X$$ simplicial boundary (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of$$\partial _\triangle X$$ to define a simplicial Roller boundary$${\partial _R}X$$ . Then, we show that$${\mathfrak {R}}_\triangle X$$ ,$$\partial _TX$$ , and$$\partial _\triangle X$$ are all homotopy equivalent,$${\mathfrak {R}}_\triangle X$$ -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.$$\text {Aut}(X)$$