Conjecture 2.1 — Let Ω∗(Hom(π,G )/G) be the de Rham algebra consisting of all measurable differential forms on Hom(π,G )/G.

Definitions and mathematical statement

Put $X=\operatorname{Hom}(\pi,G)/G$, with conjugation quotient as in the source. In the usual surface-group setting, $\pi=\pi_1(\Sigma)$. Let $\mathscr B$ denote the source's admissible bilinear forms $B$, with associated symplectic forms $\omega_B$. The standard smooth-locus construction pairs surface-group cohomology classes by cup product followed by $B$ and evaluation on $[\Sigma]$; the exact hypotheses on $G,B$ and the smooth stratum are absent.

The conjecture says that the $\operatorname{Mod}_\Sigma$-invariant subalgebra of measurable differential forms on $X$ is generated by these $\omega_B$: \[ \Omega^*_{\mathrm{meas}}(X)^{\operatorname{Mod}_\Sigma} =\text{the unital graded wedge subalgebra generated by } \{\omega_B:B\in\mathscr B\}. \] Thus invariants in all degrees, including invariant measurable functions in degree zero, are part of the statement. Whether one fixes a connected component or allows componentwise coefficients matters and is not specified. Measurable forms alone do not supply an ordinary de Rham differential without further regularity or weak-derivative conventions; the source's phrase de Rham algebra needs clarification. Equality and invariance pointwise versus almost everywhere are also unstated.

The continuation uses a finite measure $\mu$ on $X$ and $\mathcal H=L^2(X,\mu)$, describes the mapping-class representation as unitary, and reports Andersen's communication that $H^1(\operatorname{Mod}_\Sigma;\mathcal H)=0$. Unitarity of the usual pullback representation requires $\mu$-invariance, not finiteness alone; the missing measure definition must provide it. A further conjecture generalizing this one is announced but not included, and no higher-cohomology assertion is reconstructed.

Short English statement

The source conjectures that invariant measurable differential forms on the representation quotient are generated by its bilinear-form symplectic structures, with unresolved regularity, component and measure conventions.

Sources

• "Mapping class group dynamics on surface group representations," in Problems on Mapping Class Groups and Related Topics, edited by Farb, Conjecture 2.1 and continuation, supplied PDF page 215. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. The preceding definitions of $G,\pi,B,\mu$ are absent.
• Andersen, source-reported communication claiming the first-cohomology vanishing; no publication reference or complete attribution is supplied. The announced following conjecture is missing.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000186$, exact source code catalogue entry; supplied OPEN-TRIAGE summary says truncated.