Problem 2.2 — How is the Nielsen-Thurston trichotomy related to the question of whether the distance is 0, 1, 2 or≥ 3?

Definitions and mathematical statement

For a mapping class $\varphi$ of a suitable finite-type surface, the Nielsen--Thurston classification distinguishes periodic, reducible nonperiodic, and pseudo-Anosov types. Periodic means finite order; reducibility involves preservation of an essential multicurve; the precise surface and exceptional-case conventions are not specified here. Retain $d(\varphi)$ as the source's nonnegative distance, whose definition is absent.

Problem 2.2. Determine how the Nielsen--Thurston type of $\varphi$ is related to the four possibilities \[ d(\varphi)=0,\qquad d(\varphi)=1,\qquad d(\varphi)=2,\qquad d(\varphi)\geq3. \] The request is for the relationship between these classifications, including whatever implications or realizability restrictions hold in the intended setting. It does not assert a one-to-one correspondence between the three mapping-class types and the four distance cases.

The research summary suggests Hempel distance. In such an interpretation, one must first fix the Heegaard surface and disk sets $\mathcal D_1,\mathcal D_2$ and the gluing convention, so that a possible distance is $d_{\mathcal C(S)}(\mathcal D_1,\varphi(\mathcal D_2))$ in the curve graph. This is an explanatory conditional interpretation, not a definition supplied by the excerpt. It must not be replaced by the point displacement or stable translation length of $\varphi$, which are different quantities. The source's trailing sentence says that the next three problems address the nonconstructive definition of $d(\varphi)$; those problems and the definition are not included.

Short English statement

Relate the mapping class's periodic/reducible/pseudo-Anosov type to whether the source distance is $0,1,2$, or at least $3$. The distance and its auxiliary Heegaard data are missing, and a curve-graph translation length cannot be silently substituted.

Sources

• Local UnsolvedMath registry, permanent ID 11000159, source code catalogue entry, Problem 2.2, PDF page 156. No individual proposer is supplied; page distinguishes this label from repeated Problem 2.2 headings.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The volume contains 329 explicitly typeset headings across contributed chapters.
• The supplied research summary describes the excerpt as truncated and related to Hempel distance and mapping-class type, but gives no definition, theorem, or additional reference.
• Supplied status open and OPEN-TRIAGE. The summary says it cannot verify the question. Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.