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Why CAT(0) cube complexes should be replaced with median graphs

Anthony Genevois

TL;DR

The paper argues for replacing the standard terminology '$CAT(0)$ cube complexes' with 'median graphs' in geometric group theory, contending that the essential geometry is captured by median-graph structure and hyperplane combinatorics. It demonstrates that median geometry yields natural tools (e.g., Roller boundary, translation lengths via singularities) and provides a robust framework for understanding group actions, fixed-point phenomena, and Tits-type alternatives, with concrete instances such as $\mathrm{PDiff}(\mathbb{S}^1)$ and Cremona groups. While acknowledging occasional advantages of $CAT(0)$ geometry (notably in robotics and certain contexts), it argues for a nuanced program that foregrounds median graphs and their cube-completions, facilitating more computable and conceptually transparent analyses. Overall, the work promotes a median-graph perspective as a natural, efficient, and conceptually rich language for the study of groups acting on cube-like spaces, while respecting historical roots and the occasional utility of nonpositive curvature.

Abstract

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.

Why CAT(0) cube complexes should be replaced with median graphs

TL;DR

The paper argues for replacing the standard terminology ' cube complexes' with 'median graphs' in geometric group theory, contending that the essential geometry is captured by median-graph structure and hyperplane combinatorics. It demonstrates that median geometry yields natural tools (e.g., Roller boundary, translation lengths via singularities) and provides a robust framework for understanding group actions, fixed-point phenomena, and Tits-type alternatives, with concrete instances such as and Cremona groups. While acknowledging occasional advantages of geometry (notably in robotics and certain contexts), it argues for a nuanced program that foregrounds median graphs and their cube-completions, facilitating more computable and conceptually transparent analyses. Overall, the work promotes a median-graph perspective as a natural, efficient, and conceptually rich language for the study of groups acting on cube-like spaces, while respecting historical roots and the occasional utility of nonpositive curvature.

Abstract

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.
Paper Structure (8 sections, 3 theorems, 6 equations)

This paper contains 8 sections, 3 theorems, 6 equations.

Table of Contents

  1. CAT(0) geometry is not necessary.
  2. CAT(0) geometry not easily accessible.
  3. Median geometry may be enlightening.
  4. CAT(0) geometry may be inefficient.
  5. Median geometry is conceptually important.
  6. Historical background.
  7. Median graphs or median cube complexes?
  8. CAT(0) geometry somewhere?

Key Result

Theorem 1

A cube complex is CAT(0) if and only if it is simply connected and all its vertices have flag simplicial links.

Theorems & Definitions (3)

  • Theorem 1: MR0919829MR3029427
  • Theorem 2: MR1663779MR1748966Roller
  • Theorem 3