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Survey on invariant quasimorphisms and stable mixed commutator length

Morimichi Kawasaki, Mitsuaki Kimura, Shuhei Maruyama, Takahiro Matsushita, Masato Mimura

Abstract

A homogeneous quasimorphism $φ$ on a normal subgroup $N$ of $G$ is said to be $G$-invariant if $φ(gxg^{-1}) = φ(x)$ for every $g \in G$ and for every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic geometry and the extension problem of quasimorphisms. Moreover, it is known that the existence of non-extendable invariant quasimorphisms is closely related to the behavior of the stable mixed commutator length $\mathrm{scl}_{G,N}$, which is a certain generalization of the stable commutator length $\mathrm{scl}_G$. In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length. The topics we treat include several examples of invariant quasimorphisms, Bavard's duality theorem for invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of the dimension of spaces of non-extendable quasimorphisms. We also mention the extension problem of partial quasimorphisms.

Survey on invariant quasimorphisms and stable mixed commutator length

Abstract

A homogeneous quasimorphism on a normal subgroup of is said to be -invariant if for every and for every . Invariant quasimorphisms have naturally appeared in symplectic geometry and the extension problem of quasimorphisms. Moreover, it is known that the existence of non-extendable invariant quasimorphisms is closely related to the behavior of the stable mixed commutator length , which is a certain generalization of the stable commutator length . In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length. The topics we treat include several examples of invariant quasimorphisms, Bavard's duality theorem for invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of the dimension of spaces of non-extendable quasimorphisms. We also mention the extension problem of partial quasimorphisms.
Paper Structure (21 sections, 37 theorems, 66 equations, 2 figures)

This paper contains 21 sections, 37 theorems, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quasimorphisms and commutator length
  3. Invariant quasimorphisms and mixed commutator length
  4. Extension problem of quasimorphisms
  5. Comparison problem of $\mathop{\mathrm{\mathrm{scl}}}\nolimits_G$ and $\mathop{\mathrm{\mathrm{scl}}}\nolimits_{G,N}$
  6. Prehistory of invariant quasimorphisms in symplectic geometry
  7. History of invariant quasimorphisms
  8. Explicit construction of invariant quasimorphisms
  9. Py's Calabi quasimorphism
  10. Entov--Polterovich's partial Calabi quasimorphism
  11. Invariant quasimorphisms for groups acting on the circle
  12. Aut-invariant quasimorphisms
  13. On the proof of invariant Bavard's duality
  14. $\nu_N$-quasimorphism
  15. Filling norm
  16. ...and 6 more sections

Key Result

Theorem 1.1

For every $x \in [G,G]$, Here we set the supremum in the right-hand side of the above equality to be zero if $\mathrm{Q}(G) = \mathrm{H}^1(G)$.

Figures (2)

  • Figure 1: Bavard duality
  • Figure 2:

Theorems & Definitions (61)

  • Theorem 1.1: Bav
  • Theorem 1.2: KK,KKMM1
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Example 2.1
  • Example 2.2
  • Example 2.3: EP09
  • Example 2.4: Ka19
  • Example 3.1: Sh
  • ...and 51 more