Hofer-Lipschitz Quasimorphisms

• Hofer-Lipschitz quasimorphisms are real-valued functions on Hamiltonian diffeomorphism groups that satisfy a bounded defect property and a Lipschitz condition relative to the Hofer norm.
• They are constructed using spectral invariants from Floer homology and persistence module theory to quantify energy, rigidity, and group dynamics.
• These quasimorphisms have applications in detecting undistorted elements, establishing quasi-isometric embeddings, and gaining insights into bounded cohomology, with concrete examples on genus-zero surfaces and toric degenerations.

A Hofer-Lipschitz quasimorphism is a real-valued function on a group—typically the group of Hamiltonian diffeomorphisms of a symplectic manifold—that is both a quasimorphism (i.e., deviates from a homomorphism by a uniformly bounded defect) and satisfies a Lipschitz condition with respect to the Hofer norm. Such quasimorphisms play a vital role in symplectic topology, Hamiltonian dynamics, and large-scale geometry of transformation groups, connecting deep structural properties (such as bounded cohomology, rigidity, and stable commutator length) to symplectic invariants and persistence-theoretic constructions.

1. Definition and Core Properties

Given a symplectic manifold $(M, \omega)$, the Hofer norm on the group $\operatorname{Ham}(M, \omega)$ is defined via normalized Hamiltonians $H \in C^\infty_c([0,1]\times M,\mathbb{R})$ as: $$\| H \| = \int_0^1 \left( \max_M H_t - \min_M H_t \right) dt,$$ with the Hofer norm of a Hamiltonian diffeomorphism $\varphi$ being the infimum of $\|H\|$ over all $H$ generating $\varphi$. A homogeneous quasimorphism $r\colon \operatorname{Ham}(M,\omega)\to\mathbb{R}$ is Hofer-Lipschitz if there exists a constant $C>0$ such that

$|r(\varphi)| \leq C \cdot \|\varphi\|_{H}$

for all $\varphi \in \operatorname{Ham}(M,\omega)$. The quasimorphism property imposes a defect bound: $$\sup_{\varphi,\psi} |r(\varphi\psi) - r(\varphi) - r(\psi)| < \infty.$$ Homogeneous quasimorphisms are required to satisfy $r(\varphi^k) = k r(\varphi)$ for all integers $k$.

2. Floer and Persistence Module Constructions

The contemporary construction of many Hofer-Lipschitz quasimorphisms relies on filtered Floer homology, spectral invariants, and persistence module theory. A key mechanism is to associate to a Hamiltonian diffeomorphism $\varphi$ a spectral invariant $c(\varphi)$ (often defined via Floer or Morse-theoretic filtrations), and then homogenize: $$\mu(\varphi) = \lim_{k\to\infty} \frac{c(\varphi^k)}{k}.$$ Such limits yield homogeneous quasimorphisms. The spectral invariants themselves are 1-Lipschitz with respect to Hofer's metric, relying on action filtration subadditivity and persistence module stability. For example, $$|c(\varphi) - c(\psi)| \leq d_\mathrm{Hofer}(\varphi, \psi)$$, so the homogenized $\mu$ inherits the Hofer-Lipschitz property (Monzner et al., 2011, Khanevsky, 2014, Kawamoto, 2020, Kawamoto, 2022).

Persistence modules with operator structures, as in Floer theory, provide the algebraic framework for tensoring (Künneth type) constructions and for tracking the action of formal Novikov parameters on filtered chain complexes. These perspectives clarify and reinforce the filtration and operator properties responsible for Lipschitz continuity and homogeneity in the constructed quasimorphisms.

3. Applications and Examples

a) Genus-Zero Surfaces and Calabi Quasimorphisms

On $S^2$ and genus-zero surfaces, the Entov–Polterovich Calabi quasimorphism is the prototypical Hofer-Lipschitz quasimorphism, defined via spectral invariants associated to the median of the Reeb graph. For Hamiltonian diffeomorphisms $\varphi$, the Calabi quasimorphism satisfies $|\mathrm{Cal}_{S^2}(\varphi)| \leq \|\varphi\|$ (Khanevsky, 2014, Khanevsky, 2019). Its pullbacks yield families of Hofer-Lipschitz quasimorphisms with explicit homological meaning (e.g., measuring winding numbers of periodic disks).

b) Toric Degeneration and Quantum Cohomology

For higher-dimensional targets such as quadric hypersurfaces and del Pezzo surfaces, toric degeneration reveals canonical Lagrangian submanifolds (e.g., monotone tori and vanishing cycles). The corresponding quantum cohomology rings decompose into semisimple summands with idempotents $e_+$, $e_-$, and the associated spectral invariants produce distinct Hofer-Lipschitz quasimorphisms

$$\mu_{e}(\varphi) = \lim_{k \to \infty}\frac{c(\varphi^k, e)}{k}$$

(Kawamoto, 2022, Kawamoto, 2020). The superheaviness of the canonical Lagrangians with respect to different idempotents underpins the distinctness and geometric significance of such quasimorphisms.

c) Quasi-Isometric Embeddings and Growth in Hofer's Metric

Hofer-Lipschitz quasimorphisms underlie the existence of quasi-isometric embeddings of finite- and infinite-dimensional normed spaces (e.g., $[0,1]^\infty$) into $\operatorname{Ham}(M, \omega)$ equipped with the Hofer metric (Stevenson, 2016, Usher, 2013). The construction typically encodes sequence data into autonomous Hamiltonians with localized supports, so that both the Hofer norm and the quasimorphism values scale linearly with the $\ell^\infty$-distance between parameter sequences.

4. Algebraic and Geometric Implications

a) Bounded Cohomology and Rigidity

The existence of Hofer-Lipschitz quasimorphisms results in nontrivial classes in second bounded cohomology, which are sensitive to the geometry of the group and its automorphisms (Minasyan et al., 4 Sep 2025). The automorphism group acts on the space of homogeneous quasimorphisms, and the invariant structure detects inner vs. outer automorphism distinctions through the failure of strong commensuration. This has consequences for the faithfulness of the action of outer automorphism groups on bounded cohomology and on spaces of "coarse automorphisms."

b) Detection of Undistorted Elements

Hofer-Lipschitz quasimorphisms provide a test for undistorted elements: in any group with a bi-invariant norm (such as the Hofer metric), every element with linear norm growth is detected by an antisymmetric homogeneous partial quasimorphism which is Lipschitz (Kędra, 2022). The general homogenization procedure ensures that such structures can always be constructed from bounded-on-generators partial quasimorphisms.

5. Behavior, Pathologies, and Limitations

While the existence of Hofer-Lipschitz quasimorphisms is generic in some settings (e.g., for spectral or Calabi-type invariants), many classical constructions—such as Polterovich or Gambaudo–Ghys quasimorphisms based on topological or braid data—are not Hofer-Lipschitz: they often fail to be continuous or Lipschitz with respect to the Hofer metric, sometimes exhibiting arbitrarily large quasimorphism value with bounded Hofer energy (Khanevsky, 2019). This precludes their use as robust energy estimators in Hofer geometry.

On the other hand, every Hofer-Lipschitz function (satisfying appropriate partial quasimorphism inequalities) can be homogenized, and the existence of such a quasimorphism ensures the possibility of detecting unbounded metric behavior and constructing large-scale geometric embeddings.

6. Hofer-Lipschitz Constants and Structural Estimates

The Lipschitz constant for the extension-inclusion map

$$\operatorname{Ham}_c(U) \to \operatorname{Ham}_c(M)$$

gives rise to the asymptotic Hofer-Lipschitz constant (Khanevsky et al., 2011): $$\operatorname{Lip}^\infty(M, U) = \lim_{C \to \infty} \sup\left\{ \frac{\|\varphi\|_{M}}{\|\varphi\|_{U}} : \varphi \in \operatorname{Ham}_c(U),\, \|\varphi\|_{U} > C \right\},$$ which can often be computed exactly in certain product settings. This reflects the "energy increase" under trivial extension and, in many models, can be made arbitrarily small as $U$ shrinks in $M$, quantifying the breakdown of isometry in Hamiltonian extension.

7. Interplay with Bounded Cohomology, Dynamics, and Persistence

Hofer-Lipschitz quasimorphisms have connections to dynamical invariants (rotation numbers, translation numbers in circle actions), Aubry–Mather theory, weak KAM theory, and symplectic rigidity phenomena. In contexts as diverse as acylindrically hyperbolic groups or Hamiltonian group actions on configuration spaces, the algebraic structure of quasimorphisms interacts with group dynamics through cocycle and boundary constructions (Björklund et al., 2023, Maruyama et al., 2022). In the symplectic setting, the dynamical persistence encoded via spectral and persistence module invariants ensures both stability (under $C^0$-limits) and the existence of nontrivial Lipschitz quasimorphisms on transformation groups.

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In summary, Hofer-Lipschitz quasimorphisms are structurally significant objects bridging symplectic topology, group actions, dynamical systems, and large-scale geometry. Their existence and explicit construction (generally via homogenized spectral invariants and persistence-theoretic techniques) are now central tools for capturing energy growth, rigidity, and group-theoretic complexity in Hamiltonian dynamics and beyond. The development of their theory continues to inform advances in the paper of transformation groups, symplectic invariants, and quantum cohomological structures.