Homotopy equivalent boundaries of cube complexes

Talia Fernós, David Futer, Mark Hagen*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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 ∂RX (which depends only on the combinatorial structure), and the simplicial boundary ∂ΔX (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ∂RX to define a simplicial Roller boundary RΔX. Then, we show that ∂T X, ∂ΔX, and RΔX 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.
Original languageEnglish
Article number33
Number of pages83
JournalGeometriae Dedicata
Issue number2
Early online date27 Jan 2024
Publication statusPublished - 1 Apr 2024

Bibliographical note

Funding Information:
We are grateful to Craig Guilbault, Jingyin Huang, Dan Ramras, and Kim Ruane for some helpful discussions. In particular, we thank Ramras for a correction, for pointing out the reference [Bjo81], and for the observation about Borel homotopy equivalence (Remark 1.2). We thank the organizers of the conference “Nonpositively curved groups on the Mediterranean” in May of 2018, where the three of us began collaborating as a unit. Fernós was partially supported by NSF grant DMS-005640. Futer was partially supported by NSF grant DMS-907708. Hagen was partially supported by EPSRC New Investigator Award EP/R042187/1.

Publisher Copyright:
© 2024, The Author(s).


  • math.GT
  • math.GR
  • math.MG
  • 51F99 (Primary) 20F65 (Secondary)


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