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  1. Abstract

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

     
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  2. null (Ed.)