Homotopy equivalent boundaries of cube complexes

Talia Fernós; David Futer; Mark Hagen

doi:10.1007/s10711-023-00877-w

Homotopy equivalent boundaries of cube complexes

Talia Fernós, David Futer, Mark Hagen^*

^*Corresponding author for this work

School of Mathematics

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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 ∂_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 R_ΔX. Then, we show that ∂_T X, ∂_ΔX, and R_ΔX 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 language	English
Article number	33
Number of pages	83
Journal	Geometriae Dedicata
Volume	218
Issue number	2
Early online date	27 Jan 2024
DOIs	https://doi.org/10.1007/s10711-023-00877-w
Publication status	Published - 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).

Keywords

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

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New Implications of Arboreality and Hyperbolicity for Groups
Hagen, M. F. (Principal Investigator)
1/08/18 → 28/02/21
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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 ∂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.",

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N2 - 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.

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Homotopy equivalent boundaries of cube complexes

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