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Balanced curves, quasimorphisms, and the equator conjecture

Yongsheng Jia, Richard Webb

TL;DR

The paper develops a new, infinite-dimensional family of homogeneous quasimorphisms on Ham($S^2$) and $ ext{Homeo}_0(S^2,oldsymbol{ m \omega})$ by acting on a novel hyperbolic graph of balanced curves. Central to the construction is the $oldsymbol{ rak{C}}^{ }_{oldsymbol{ ightarrow}oldsymbol{ earrow}}(S^{2})$ graph, which is connected, of infinite diameter, and Gromov-hyperbolic, with a projection framework to subsurfaces that enables Bestvina–Fujiwara quasimorphisms. By engineering independent hyperbolic elements via small-area subsurfaces (witnesses) and exploiting coarse projections, the authors obtain an infinite-dimensional space of $C^0$-continuous, Hofer-Lipschitz quasimorphisms vanishing on stabilizers of $oldsymbol{ rak{a}}$-balanced curves, including equators. As an application, they prove that the space of equators under the $A$-fragmentation metric has infinite diameter for any $0<A<1$, and discuss connections to the equator conjecture through Hofer-dynamics. The results combine curve-graph techniques with symplectic topology, yielding a robust BF-quasimorphism toolkit for $ ext{Ham}(S^2)$ and its measure-preserving homeomorphism variant.

Abstract

We construct a new infinite-dimensional family of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere. Moreover, for any constant $K$ less than the total area of the sphere, we produce unbounded homogeneous quasimorphisms that vanish on any map supported on some disk of area at most $K$. As an application, we prove an analogue of the equator conjecture, namely that the space of equators equipped with any choice of quantitative fragmentation metric has infinite diameter. To prove our results, we introduce tools that draw inspiration from the theory of curve graphs used to study mapping class groups.

Balanced curves, quasimorphisms, and the equator conjecture

TL;DR

The paper develops a new, infinite-dimensional family of homogeneous quasimorphisms on Ham() and by acting on a novel hyperbolic graph of balanced curves. Central to the construction is the graph, which is connected, of infinite diameter, and Gromov-hyperbolic, with a projection framework to subsurfaces that enables Bestvina–Fujiwara quasimorphisms. By engineering independent hyperbolic elements via small-area subsurfaces (witnesses) and exploiting coarse projections, the authors obtain an infinite-dimensional space of -continuous, Hofer-Lipschitz quasimorphisms vanishing on stabilizers of -balanced curves, including equators. As an application, they prove that the space of equators under the -fragmentation metric has infinite diameter for any , and discuss connections to the equator conjecture through Hofer-dynamics. The results combine curve-graph techniques with symplectic topology, yielding a robust BF-quasimorphism toolkit for and its measure-preserving homeomorphism variant.

Abstract

We construct a new infinite-dimensional family of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere. Moreover, for any constant less than the total area of the sphere, we produce unbounded homogeneous quasimorphisms that vanish on any map supported on some disk of area at most . As an application, we prove an analogue of the equator conjecture, namely that the space of equators equipped with any choice of quantitative fragmentation metric has infinite diameter. To prove our results, we introduce tools that draw inspiration from the theory of curve graphs used to study mapping class groups.

Paper Structure

This paper contains 21 sections, 30 theorems, 29 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs as metric spaces
  4. Graphs as metric spaces
  5. Gromov hyperbolicity
  6. Quasi-isometry
  7. Arc and curve graphs
  8. Intersections and dual trees
  9. The balanced curve graph
  10. Definition and theorem
  11. Witnesses and subsurface projections
  12. Questions related to quasi-isometries
  13. Hyperbolicity of the balanced curve graph
  14. Projections and 'Guessing Geodesics'
  15. Criteria to ensure small distances
  16. ...and 6 more sections

Key Result

Theorem 1.1

For every $0<A<1$, the space of equators on $S^2$ equipped with the $A$-quantitative fragmentation metric has infinite diameter.

Figures (10)

  • Figure 1: The dual tree
  • Figure 2: Two $\varepsilon$-balanced curves with only two non-bigon complementary regions
  • Figure 3: The construction of the $\varepsilon$-balanced curve
  • Figure 4: Case analysis
  • Figure 5: The choice of $\gamma$
  • ...and 5 more figures

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 71 more