Problem 3.2 — Find general conditions which ensure that (10) is proper.

Definitions and mathematical statement

The principal problem is to find general sufficient conditions for the object denoted (10) in the source to be proper. Equation (10), its domain, codomain and nature (map or action) are absent. For a continuous map, proper means compact inverse images of compact sets; properness of a group action is a distinct condition on its action map. No particular representation action is substituted for the missing (10).

The supplied following examples use an undefined trace parameter $\kappa$ and mapping class group $\operatorname{Mod}_\Sigma$. They report the following dynamics, with source measure and character-locus conventions still to be fixed.

1. On ${\mathbb{R}}^3\cap\kappa^{-1}(2)$, characters are described as abelian, and the action is ergodic on each of four smooth connected components. For $2<t\leq18$, the action on ${\mathbb{R}}^3\cap\kappa^{-1}(t)$ is ergodic.
2. For $t>18$, proper and chaotic regions coexist. The region $( -\infty,-2]^3$ consists of characters of discrete embeddings $\rho$ with $\mathbb H^2/\rho(\pi)$ homeomorphic to a three-holed sphere. Homotopy equivalences from $\Sigma$ to such a hyperbolic surface determine these characters and closed triangular regions, described as freely permuted by $\operatorname{Mod}_\Sigma$. The complement of the wandering domains is reported ergodic.
3. For $G=\operatorname{PGL}(2,{\mathbb{R}})$, including orientation-reversing hyperbolic isometries, the source uses the double cover \[ G^\pm=\operatorname{SL}(2,{\mathbb{C}})\cap (\operatorname{GL}(2,{\mathbb{R}})\cup i\operatorname{GL}(2,{\mathbb{R}})). \] It reports four trace-coordinate components: the preceding real one and ${\mathbb{R}}\times i{\mathbb{R}}\times i{\mathbb{R}}$, $i{\mathbb{R}}\times{\mathbb{R}}\times i{\mathbb{R}}$, $i{\mathbb{R}}\times i{\mathbb{R}}\times{\mathbb{R}}$. On the indicated $i{\mathbb{R}}\times{\mathbb{R}}\times i{\mathbb{R}}$ component, it reports ergodicity for $-14\leq t<2$. For $t<-14$, wandering domains correspond to homotopy equivalences from $\Sigma$ to a hyperbolic two-holed projective plane, with ergodicity on their complement.

These are accompanying source reports, not new general properness criteria. The definitions of $\Sigma,\pi,\kappa$, exact admissible trace loci, ergodic measures, effective group action and wandering-domain boundaries are not provided; no assertion is added for omitted parameter ranges.

Short English statement

Find criteria for properness of the missing object (10), retaining the supplied real and mixed-real trace-level dynamics, thresholds and wandering-domain descriptions as context.

Sources

• "Mapping class group dynamics on surface group representations," in Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 3.2 and following examples, supplied PDF page 221. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. Equation (10) and the preceding trace definitions are absent.
• Stantchev [94, 52], source attribution for the beginning of the $\operatorname{PGL}(2,{\mathbb{R}})$ analysis; full bibliography entries are not supplied.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000195$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports unverified status.