Conjecture 3.2 — If k = 1, then U is onto.

Definitions and mathematical statement

Conjecture 3.2 has two assertions concerning the source's map $U$ and parameter $k$:

1. If $k=1$, then $U$ is surjective.
2. In general, every $\operatorname{PSL}_2({\mathbb{R}})$ representation with dense image lies in $\operatorname{Image}(U)$.

Here $\operatorname{PSL}_2({\mathbb{R}})=\operatorname{SL}_2({\mathbb{R}})/\{\pm I\}$, and dense image means density in its ordinary Lie-group topology: for an intended source group $\Pi$, $\overline{\rho(\Pi)}=\operatorname{PSL}_2({\mathbb{R}})$. Zariski density is not substituted.

Neither $k$, the source group $\Pi$, nor the domain and codomain of $U$ is supplied. If the construction depends on $k$ and is written conditionally as $U_k:X_k\to Y_k$, the first claim is $U_1(X_1)=Y_1$; the second says that every admissible dense-image representation in $Y_k$ has a preimage under $U_k$, for the intended range of parameters. If $Y_k$ consists of conjugacy classes, the latter must concern $[\rho]$ rather than a chosen representative. These are only logical restatements of the two claims; the missing construction and admissibility conditions prevent a complete standalone conjecture.

Short English statement

The source conjectures surjectivity of $U$ for $k=1$ and, more generally, inclusion of all admissible dense-image $\operatorname{PSL}_2({\mathbb{R}})$ representations in its image. The map and parameters are undefined here.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000199$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Conjecture 3.2, PDF page 226, chapter identified in the extracted footer as W. Goldman: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; repeated local numbering disambiguated by page among 329 headings.
• No definition of $U$, preceding setup or additional primary citation is supplied. The registry literature summary is truncated.