Foliations and Diffeomorphism Groups — Question 30 (Calegari)

Definitions and mathematical statement

The supplied question asks whether analytic actions of surface groups on ${\mathbb{R}}^2$ satisfy the Milnor--Wood inequality. It does not explicitly specify the surfaces, the orientation convention, or the numerical inequality intended.

A conditional formulation is as follows. Let $\Sigma_g$ be a closed, connected, oriented surface of genus $g\geq 2$, and let \[ \rho\colon\pi_1(\Sigma_g)\longrightarrow\operatorname{Diff}^{\omega}_{+}({\mathbb{R}}^2) \] be a group homomorphism, where $\operatorname{Diff}^{\omega}_{+}({\mathbb{R}}^2)$ denotes the group of orientation-preserving real-analytic diffeomorphisms of the plane. No faithfulness assumption is imposed in this conditional formulation. Form the associated flat plane bundle \[ p\colon E_\rho=(\widetilde{\Sigma}_g\times{\mathbb{R}}^2)/\pi_1(\Sigma_g)\longrightarrow\Sigma_g, \qquad \gamma\cdot(x,v)=(\gamma x,\rho(\gamma)(v)). \] Since its fiber is contractible, this bundle admits a smooth section $s$. Its vertical tangent bundle $T_vE_\rho=\ker({\,\mathrm{d}} p)$ is an oriented real vector bundle of rank two. Define the Euler number of the action by \[ \operatorname{eu}(\rho)=\left\langle e\left(s^*T_vE_\rho\right),[\Sigma_g]\right\rangle\in{\mathbb{Z}}. \] This integer is independent of the section.

For a specified bound $B(g)$, the precise assertion to test would be \[ \forall g\geq 2\;\forall\rho\colon\pi_1(\Sigma_g)\to\operatorname{Diff}^{\omega}_{+}({\mathbb{R}}^2), \qquad |\operatorname{eu}(\rho)|\leq B(g). \] The source does not give $B(g)$. In particular, it does not resolve whether the intended plane-bundle bound is $B(g)=g-1$, or whether the intended bound is $B(g)=2g-2=|\chi(\Sigma_g)|$, as in the circle-action version of the Milnor--Wood inequality. These are distinct candidate assertions, and neither is selected here as the definitive source statement. The closed-surface, genus, real-analytic, and orientation-preserving conventions above also require confirmation against the original question.

Short English statement

Calegari's question, as recorded by Nariman and Yazdi, asks whether real-analytic actions of surface groups on the plane obey a Milnor--Wood bound on their Euler numbers. The supplied record does not specify the numerical bound or all surface and orientation hypotheses, so the formulation above remains conditional. The registry labels the question open; that label is supplied metadata.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800089$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 30 (Calegari)." The record lists Nariman and Yazdi as proposers, gives the year $2025$, and labels the problem open.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 30. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on $2026\text{-}10\text{-}16$ and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source verification.
• The supplied background attributes the question to Calegari, and the statement refers to his paper concerning the Euler class and planar group actions. No title, bibliographic details, or separate URL for that paper is supplied.