Constructing metric spaces from systems of walls

2024/04/18 by Harry Petyt, Petyt, Harry, Abdul Zalloum +3 · 3 citations
Engineering · #Architecture and Computational Design #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2404.12057

openalex publication_date 2024/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give a general procedure for constructing metric spaces from systems of partitions. This generalises and provides analogues of Sageev's construction of dual CAT(0) cube complexes for the settings of hyperbolic and injective metric spaces. As applications, we produce a ``universal'' hyperbolic action for groups with strongly contracting elements, and show that many groups with ``coarsely cubical'' features admit geometric actions on injective metric spaces. In an appendix with Davide Spriano, we show that a large class of groups have an infinite-dimensional space of quasimorphisms.

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Related

• Injectivity, cubical approximations and equivariant wall structures beyond CAT(0) cube complexes
• From wall spaces to CAT(0) cube complexes
• Injective hulls of certain discrete metric spaces and groups
• Hyperbolic spaces that detect all strongly-contracting directions
• Approximating hyperbolic lattices by cubulations