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Survey on Cremona groups from a median geometric point of view

Anne Lonjou

TL;DR

The survey surveys how Cremona groups act on median graphs and related CAT(0) cube complexes, bridging birational geometry with geometric group theory. It develops median-graph models (blow-up, rational blow-up, Jonquières graphs) for groups of birational transformations of surfaces, and extends these ideas to higher dimensions via C^ℓ(X) graphs, enabling regularization results and structure theorems. Core contributions include linking elliptic/purely elliptic actions to regularization, proving non-simplicity and Tits-alternative-type phenomena for rank-2 Cremona groups, and constructing morphisms from Cremona groups to Z and to right-angled Artin groups via median-graph actions. The approach provides a flexible, combinatorial toolkit to study large, non-locally compact birational groups, with open questions about fixed points in infinite-dimensional settings and higher-dimensional dynamics having significant implications for understanding birational symmetries.

Abstract

This expository article builds on lecture notes from a minicourse entitled "Cremona groups and CAT(0) cube complexes" and given by the author as part of the 2023 Riverside Workshop on Geometric Group Theory. It presents recent constructions of actions of Cremona groups on median graphs aimed at both geometric group theorists and algebraic geometers.

Survey on Cremona groups from a median geometric point of view

TL;DR

The survey surveys how Cremona groups act on median graphs and related CAT(0) cube complexes, bridging birational geometry with geometric group theory. It develops median-graph models (blow-up, rational blow-up, Jonquières graphs) for groups of birational transformations of surfaces, and extends these ideas to higher dimensions via C^ℓ(X) graphs, enabling regularization results and structure theorems. Core contributions include linking elliptic/purely elliptic actions to regularization, proving non-simplicity and Tits-alternative-type phenomena for rank-2 Cremona groups, and constructing morphisms from Cremona groups to Z and to right-angled Artin groups via median-graph actions. The approach provides a flexible, combinatorial toolkit to study large, non-locally compact birational groups, with open questions about fixed points in infinite-dimensional settings and higher-dimensional dynamics having significant implications for understanding birational symmetries.

Abstract

This expository article builds on lecture notes from a minicourse entitled "Cremona groups and CAT(0) cube complexes" and given by the author as part of the 2023 Riverside Workshop on Geometric Group Theory. It presents recent constructions of actions of Cremona groups on median graphs aimed at both geometric group theorists and algebraic geometers.

Paper Structure

This paper contains 44 sections, 61 theorems, 48 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Median geometry
  3. CAT(0) cube complexes vs median graphs
  4. Cube complexes
  5. CAT(0) metric spaces
  6. Combinatorial point of view on cube complexes
  7. Median graphs
  8. Hyperplanes
  9. Isometries
  10. Purely elliptic actions on median graphs
  11. Finite dimensional case
  12. Case of locally finite dimensional median graphs
  13. Case of infinite-dimensional median graphs not locally of finite dimension
  14. Related questions
  15. Cremona groups
  16. ...and 29 more sections

Key Result

Theorem 2.2

A cube complex is CAT(0) if and only if it is simply connected and the link of any vertex is flag.

Figures (16)

  • Figure 1: Example of a flag link.
  • Figure 2: Example of a non-flag link.
  • Figure 3: Examples of median graphs.
  • Figure 4: Example of non-median graphs.
  • Figure 5: Example of a median graph with its cubical subdivision (respectively red, orange, yellow and green vertices correspond respectively to 0,1,2,3-dimensional cubes).
  • ...and 11 more figures

Theorems & Definitions (109)

  • Definition 2.1
  • Theorem 2.2: Gromov, Leary
  • Definition 2.3
  • Theorem 2.4: Chepoi_mediangraphsGerasimov_fixed_point_free_actionroller_thesis
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Example 2.10
  • ...and 99 more