Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak, Sobhan Seyfaddini, Ivan Smith

TL;DR

This work analyzes link spectral invariants on compact surfaces, establishing a two-term Weyl law that recovers the Calabi invariant in the leading term and reveals bounded subleading behavior tied to the Ruelle invariant for genus-zero surfaces. It develops a robust framework of quasimorphisms via monotone Lagrangian links and extends Calabi in infinitely many ways to the group of compactly supported area-preserving homeomorphisms, yielding new normal subgroups and nontrivial quotient structures. A central technical device is the defect control for the quasimorphisms $f_k$ and $\mu_k$, enabling precise subleading asymptotics and explicit formulas in the autonomous disc case: $\lim_{k\to\infty} \big(k\mu_k(H)-(k+1)Cal(H)\big)=-\tfrac{1}{2}Ru(H)$. The paper also proves the simplicity of the commutator group $[G,G]$ for surface homeomorphism groups, contrasting with the non-simplicity results for kernels of Calabi in $Hameo$, thereby revealing a rich and intricate normal-subgroup structure in 2D area-preserving dynamics.

Abstract

This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disc maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

TL;DR

This work analyzes link spectral invariants on compact surfaces, establishing a two-term Weyl law that recovers the Calabi invariant in the leading term and reveals bounded subleading behavior tied to the Ruelle invariant for genus-zero surfaces. It develops a robust framework of quasimorphisms via monotone Lagrangian links and extends Calabi in infinitely many ways to the group of compactly supported area-preserving homeomorphisms, yielding new normal subgroups and nontrivial quotient structures. A central technical device is the defect control for the quasimorphisms and , enabling precise subleading asymptotics and explicit formulas in the autonomous disc case: . The paper also proves the simplicity of the commutator group for surface homeomorphism groups, contrasting with the non-simplicity results for kernels of Calabi in , thereby revealing a rich and intricate normal-subgroup structure in 2D area-preserving dynamics.

Abstract

This paper continues the study of link spectral invariants on compact surfaces, introduced in our previous work and shown to satisfy a Weyl law in which they asymptotically recover the Calabi invariant. Here we study their subleading asymptotics on surfaces of genus zero. We show the subleading asymptotics are bounded for smooth time-dependent Hamiltonians, and recover the Ruelle invariant for autonomous disc maps with finitely many critical values. We deduce that the Calabi homomorphism admits infinitely many extensions to the group of compactly supported area-preserving homeomorphisms, and that the kernel of the Calabi homomorphism on the group of hameomorphisms is not simple.
Paper Structure (32 sections, 23 theorems, 103 equations, 2 figures)

This paper contains 32 sections, 23 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Area-preserving homeomorphisms of surfaces
  3. A two-term Weyl law
  4. Infinitely many extensions of Calabi and the simplicity Conjecture revisited
  5. Simplicity
  6. Themes of the proofs and outline of the paper
  7. Summary of our knowledge of the normal subgroup structure
  8. Organization of the paper
  9. Acknowledgements
  10. Preliminaries
  11. The groups, the Calabi homomorphism and the Ruelle invariant
  12. Basic notions
  13. Some normal subgroups of $G$
  14. The Ruelle invariant
  15. Monotone links, spectral invariants and quasimorphisms
  16. ...and 17 more sections

Key Result

Theorem 1.2

The following groups are not perfect: Both admit surjective group homomorphisms to $\mathbb{R}$.

Figures (2)

  • Figure 1: Two examples of Lagrangian links on $\mathbb{S}^2$ with respectively $k=4$ and $k=5$ components.
  • Figure 2: On the left: $R^{-1}(J_i)$ (the union of pink and light blue regions) contains $R^{-1}(v_i)$ (pink region) and type $T_{2,i}$ circles (blue). $R^{-1}(V_i)$ (the union of pink, light blue and light green regions) contains both type $T_{3,i}$ circles (red) and type $T_{2,i}$ circles. Type $T_1$ circles (black) are level sets outside the interior of $R^{-1}(V_i)$. On the right: we indicate a neighborhood of the vertex $v_i$ in $G$, coloured to indicate the images of the respective regions on the left.

Theorems & Definitions (55)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.6
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • Lemma 1.11: CGHMSS21, the proof of Thm. 7.6, 7.7 and Eq. (70)
  • Definition 2.1
  • Definition 2.2: Oh-Müller muller-oh
  • ...and 45 more