[Paper Review] $\hat{G}$-invariant quasimorphisms and symplectic geometry of surfaces

Morimichi Kawasaki, Mitsuaki Kimura|arXiv (Cornell University)|Nov 25, 2019

Geometric and Algebraic Topology30 references4 citations

TL;DR

This paper introduces $Ģ{G}$-invariant quasimorphisms on a normal subgroup $G$ of a group $Ģ{G}$, establishing a Bavard-type duality for $(Ģ{G},G)$-commutator length. It applies this framework to symplectic geometry, proving the non-existence of a section for the flux homomorphism on higher-genus surfaces and showing that Py’s Calabi quasimorphism and Entov–Polterovich’s partial Calabi quasimorphism are non-extendable to the full symplectomorphism group.

ABSTRACT

Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms. We show that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.

Motivation & Objective

• To develop a theory of $Ģ{G}$-invariant quasimorphisms on a normal subgroup $G$ of a group $Ģ{G}$, motivated by applications in symplectic topology and low-dimensional topology.
• To establish a duality theorem analogous to Bavard’s duality, relating $(Ģ{G},G)$-commutator length to $Ģ{G}$-invariant quasimorphisms.
• To address the extension problem of quasimorphisms, particularly in the context of symplectic groups and flux homomorphisms.
• To prove the non-existence of a section for the flux homomorphism on closed surfaces of genus $g \geq 2$, using the structure of $Ģ{G}$-invariant quasimorphisms.
• To show that Py’s Calabi quasimorphism and Entov–Polterovich’s partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms, and that Py’s quasimorphism is the unique non-extendable one in a certain class.

Proposed method

• Define $Ģ{G}$-invariant quasimorphisms as homogeneous quasimorphisms on $G$ that are invariant under conjugation by elements of $Ģ{G}$, with defect $D(\phi)$ measuring their deviation from being a homomorphism.
• Introduce the $(\u0122{G},G)$-commutator length $\mathrm{cl}_{\u0122{G},G}(x)$ for $x \in [\u0122{G},G]$, generalizing the standard commutator length to include conjugation by $\u0122{G}$.
• Prove a Bavard-type duality: $\mathrm{scl}_{\u0122{G},G}(x) = \sup_{\phi \in Q(G)^{\u0122{G}}} \frac{1}{2} \frac{|\phi(x)|}{D(\phi)}$, linking stable $(\u0122{G},G)$-commutator length to $\u0122{G}$-invariant quasimorphisms.
• Use the 7-term exact sequence in bounded cohomology to relate $Q(G)^{\u0122{G}}$ to $H^2(H;\mathbb{R})$, where $H = \u0122{G}/G$, establishing injectivity of a map $\tau_{\u0122{G}}$ from $Q(G)^{\u0122{G}}/Q(\u0122{G})$ to $H^2(H;\mathbb{R})$.
• Apply the duality and cohomological tools to symplectic manifolds, particularly surfaces, using the flux homomorphism $\mathrm{Flux}_\omega: \mathrm{Symp}_0(M,\omega) \to H^1(M;\mathbb{R})/\Gamma_\omega$.
• Leverage amenability of $H^1(M;\mathbb{R})/\Gamma_\omega$ and perfectness of $\mathrm{Ham}(M,\omega)$ to constrain the space of $\u0122{G}$-invariant quasimorphisms and prove non-extendability results.

Experimental results

Research questions

• RQ1 What is the stable $(\u0122{G},G)$-commutator length, and how does it relate to $\u0122{G}$-invariant quasimorphisms?
• RQ2 Under what conditions can a quasimorphism on $G$ be extended to $\u0122{G}$, and when is such an extension impossible?
• RQ3 Does the flux homomorphism on the symplectomorphism group of a closed surface of genus $g \geq 2$ admit a section?
• RQ4 Are Py’s Calabi quasimorphism and Entov–Polterovich’s partial Calabi quasimorphism extendable to the full symplectomorphism group?
• RQ5 Is Py’s Calabi quasimorphism the unique non-extendable quasimorphism in the space of $\u0122{G}$-invariant quasimorphisms for certain symplectic surfaces?

Key findings

• The paper establishes a Bavard-type duality for $(\u0122{G},G)$-commutator length: $\mathrm{scl}_{\u0122{G},G}(x) = \sup_{\phi \in Q(G)^{\u0122{G}}} \frac{1}{2} \frac{|\phi(x)|}{D(\phi)}$ for $x \in [\u0122{G},G]$.
• It proves the non-existence of a section for the flux homomorphism on closed surfaces of genus $g \geq 2$, using the injectivity of the map $\tau_{\u0122{G}}$ and the structure of $H^2(H;\mathbb{R})$.
• Py’s Calabi quasimorphism on $\mathrm{Ham}(\Sigma,\omega)$ is shown to be non-extendable to $\mathrm{Symp}_0(\Sigma,\omega)$, and it is the unique non-extendable quasimorphism in its class.
• Entov–Polterovich’s partial Calabi quasimorphism is also non-extendable to the symplectomorphism group, confirming its intrinsic invariance under the larger group.
• The dimension of the space $Q(\mathrm{Ham}(M,\omega))^{\u0122{G}}/Q(\u0122{G})$ is bounded by $N(N-1)/2$ when $H \cong \mathbb{Z}^N$, as shown via cohomological methods.
• For a surface $\Sigma$ of genus $g \geq 2$, the space $Q(\mathrm{Ham}(\Sigma,\omega))^{\hat{G}_0}/Q(\hat{G}_0)$ has dimension at most 1, and since $[\mu_P]$ is non-trivial, it is exactly 1, proving uniqueness of the non-extendable quasimorphism.