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Peer-reviewedCC BYMathematische Annalen

Constructing metric spaces from systems of walls

  • Harry Petyt1
  • Abdul Zalloum2
  1. 1Mathematics Institute, University of Warwick, Coventry, UK
  2. 2Department of Mathematics and Statistics, Queen’s University, Kingston, Canada
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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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Cite

BibTeXRIS
BibTeX
@article{petyt2026constructing,
  title = {{Constructing metric spaces from systems of walls}},
  author = {Harry Petyt and Abdul Zalloum},
  journal = {Mathematische Annalen},
  year = {2026},
  month = sep,
  doi = {10.1007/s00208-026-03534-1},
  url = {https://doi.org/10.1007/s00208-026-03534-1},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Constructing metric spaces from systems of walls
AU  - Harry Petyt
AU  - Abdul Zalloum
PY  - 2026
DA  - 2026/09/15
JO  - Mathematische Annalen
DO  - 10.1007/s00208-026-03534-1
UR  - https://doi.org/10.1007/s00208-026-03534-1
AB  - 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.
SN  - 0025-5831
SN  - 1432-1807
ER  -