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Homotopy equivalent boundaries of cube complexes

Fernós, Talia
; 
Futer, David
; 
Hagen, Mark
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Journal article
Date
2024-01-27
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Mathematics
Subject
CAT(0) cube complex
 Tits boundary
 Roller boundary
 Nerve theorem
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https://scholarshare.temple.edu/handle/20.500.12613/11705
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https://doi.org/10.1007/s10711-023-00877-w
Abstract
A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary ∂T X (which depends on the CAT(0) metric), the Roller boundary ∂R X (which depends only on the combinatorial structure), and the simplicial boundary ∂ X (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ∂R X to define a simplicial Roller boundary R X. Then, we show that ∂T X, ∂ X, and R X 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.
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Fernós, T., Futer, D. & Hagen, M. Homotopy equivalent boundaries of cube complexes. Geom Dedicata 218, 33 (2024). https://doi.org/10.1007/s10711-023-00877-w
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Springer Science and Business Media
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Geometriae Dedicata, Vol. 218, Iss. 2
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