Foliations and Diffeomorphism Groups — Question 17

Definitions and mathematical statement

Let \[ D^2=\{x\in{\mathbb{R}}^2:\|x\|\leq 1\},\qquad *=0, \] and, following the source's explicit description, let \[ G=\{f:D^2\longrightarrow D^2:\ f\text{ is a smooth orientation-preserving diffeomorphism and }f(0)=0\}, \] with group operation composition. No requirement that the boundary be fixed pointwise is stated. The source denotes this group by $\mathrm{Diff}_0(D^2,*)$; it does not separately explain the subscript $0$.

A real-valued quasimorphism on $G$ is a function $q:G\to{\mathbb{R}}$ with finite defect \[ D(q)=\sup_{g,h\in G}|q(gh)-q(g)-q(h)|<\infty. \] It is homogeneous if \[ q(g^n)=nq(g)\qquad(g\in G,\ n\in{\mathbb{Z}}). \] Write $\mathcal Q_{\mathrm{hom}}(G)$ for the real vector space of all such homogeneous quasimorphisms, with pointwise operations. This definition imposes no continuity or measurability condition and takes no quotient by homomorphisms or bounded functions; the source does not explicitly discuss these conventions.

The source asserts that there are two bounded Euler classes associated with this group, one induced by restriction to $\partial D^2$, and the other induced by the derivative at $0$. It further asserts that their difference gives a nontrivial quasimorphism on $G$. These are supplied background assertions; the record does not specify the class normalizations or the construction of that quasimorphism.

Question 17 (Nariman, Yazdi, 2025). Is \[ \dim_{{\mathbb{R}}}\mathcal Q_{\mathrm{hom}}(G)=1? \] This is the question under the explicit group description and quasimorphism conventions above. Any intended component restriction, regularity requirement, or quotient convention requires clarification from the source.

Short English statement

For the group of smooth orientation-preserving diffeomorphisms of the closed disk fixing its marked origin, is the real vector space of homogeneous quasimorphisms one-dimensional? The source supplies a nontrivial quasimorphism arising from two bounded Euler classes. The precise conventions for the group notation and the space of quasimorphisms require review.

Sources

• Local UnsolvedMath registry, permanent ID $800076$, exact source problem code catalogue entry, supplied complete record titled Foliations and Diffeomorphism Groups --- Question 17. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Nariman, Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 17. Supplied source URL: https://arxiv.org/abs/2503.18274. The record attributes the question to Nariman and Yazdi, with proposed year 2025.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. This access information is supplied provenance, not an independently verified source visit. The registry labels the problem open; that label is supplied curation, not an independently verified literature assessment.