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 (which depends on the CAT(0) metric), the Roller boundary (which depends only on the combinatorial structure), and the simplicial boundary (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of the Roller boundary to obtain the simplicial Roller boundary. Then, we show that the Tits, simplicial, and simplicial Roller boundaries are all homotopy equivalent, $Aut(X)$--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.

A few typos fixed and references updated. Journal version: https://link.springer.com/article/10.1007/s10711-023-00877-w

References in corpus (14)

• Rank rigidity for CAT(0) cube complexes
• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Cubulating spaces with walls
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• At infinity of finite-dimensional CAT(0) spaces
• On hierarchical hyperbolicity of cubical groups
• The simplicial boundary of a CAT(0) cube complex
• Top dimensional quasiflats in $CAT(0)$ cube complexes
• Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries
• Contracting isometries of CAT(0) cube complexes and acylindrical hyperbolicity of diagram groups
• Roller boundaries for median spaces and algebras
• $\mathrm{CAT}(0)$ cube complexes are determined by their boundary cross ratio
• Corrigendum to "The simplicial boundary of a CAT(0) cube complex''
• Deforming cubulations of hyperbolic groups