Conjecture 3.1 — Suppose that b = 0 (Σ is closed).

Definitions and mathematical statement

Assume $b=0$, so $\Sigma$ is closed of genus $g$. Let $\pi,G$ and the component invariant $e$ have their source meanings; none is defined in this extract. Conjecture 3.1 asks that for every integer \[ 1\le k\le2g-2, \] the $\operatorname{Mod}_\Sigma$-action on \[ e^{-1}(2-2g+k) \] be ergodic. This simplifies the source's $2-2g+b+k$ and $2g+b-2$ expressions using its explicit $b=0$ hypothesis, without proposing a version for nonzero $b$.

For a specified invariant measure $\mu$, ergodicity means every measurable invariant set has either measure zero or complement of measure zero. The record supplies no measure or measure class. It writes the components as subsets of $\operatorname{Hom}(\pi,G)$, yet gives the identification \[ e^{-1}(3-2g)\approx\Sigma\times{\mathbb{R}}^{6g-8}, \] a $(6g-6)$-dimensional thickening. This is consistent with the dimension of a usual surface character-variety model, rather than the unquotiented representation space; the quotient and smooth/topological meaning of $\approx$ require checking. A mapping-class action defined through outer automorphisms also needs this distinction. An Euler-class interpretation of $e$, for example for $\mathrm{PSL}_2({\mathbb{R}})$, is plausible terminology but is not a supplied choice of $G$.

The context contrasts the action on this thickening with realization on $\Sigma$ itself. It attributes to Morita [81] nonexistence of a smooth action inducing the natural projection $\operatorname{Diff}(\Sigma)\to\operatorname{Mod}_\Sigma$, with precise regularity/genus hypotheses absent. It reports a then-recent announcement by Markovic [80] excluding analogous homeomorphism sections for closed genus greater than $5$. These are source-era reports, not current verification. They do not supply the omitted measure or settle the ergodicity conjecture.

Short English statement

For a closed surface, the specified nonextremal components are conjectured ergodic under the mapping class action. The representation group, quotient and measure are missing. The source's thickening model and historical smooth/homeomorphism realization obstructions are retained as context.

Sources

• Local UnsolvedMath registry, permanent ID $11000193$, exact source code catalogue entry, Conjecture 3.1, PDF page 220. No individual proposer or proposed year is supplied; the title is truncated.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The volume has 329 explicitly typeset numbered items across chapters, repeated local numbering distinguished by PDF page.
• Internal references: Morita [81] for the reported smooth nonrealization and Markovic [80] for the source-era homeomorphism announcement in genus greater than five. Full entries, dates and exact theorem hypotheses are absent.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and calls for a current primary-literature audit. Registry status open and classification OPEN-TRIAGE; the research summary says only "Literature status: Truncated."
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.