Question 7.4 — Is there a unique (up to conjugacy) minimal dilation pseudo-Anosov in Modg?

Definitions and mathematical statement

For a pseudo-Anosov mapping class $f\in\operatorname{Mod}_g$, let $\lambda(f)>1$ be its dilatation, the expansion factor for its invariant measured foliations. Write \[ \delta_g=\inf\{\lambda(f):f\in\operatorname{Mod}_g \text{ pseudo-Anosov}\}. \] Question 7.4 asks whether the classes attaining this minimum form one conjugacy class: \[ \left|\{[f]_{\rm conj}: f\text{ pseudo-Anosov},\ \lambda(f)=\delta_g\}\right|=1\ ? \] Conjugacy is in the source's $\operatorname{Mod}_g$, not an unspecified larger equivalence allowing inversion, central multiplication or orientation reversal. The genus range and exact surface conventions are not supplied.

The source claims uniqueness for $g=1$, represented by \[ A=\begin{pmatrix}2&1\\1&1\end{pmatrix}. \] This needs qualification under the standard $\operatorname{Mod}_1\cong\mathrm{SL}_2({\mathbb{Z}})$ convention: $A$ and $-A$ have the same expansion-factor modulus but different traces, so are not conjugate in that group. Thus the stated genus-one example may be using a sign quotient or an extra foliation/action convention. It is retained as a source claim rather than used as an unqualified base case.

The subsequent context proposes refining the minimum-dilatation question by singularity data. Fix a genus and an admissible tuple $(k_1,\ldots,k_r)$ recording the prong numbers at the stable foliation's $r$ singularities. In the usual closed genus-at-least-two convention, singularities have $k_i\ge3$ and Poincaré--Hopf gives $\sum_i(k_i-2)=4g-4$, hence finitely many possible tuples. These necessary constraints are not asserted sufficient for the source's notion of admissibility. The source cites Masur--Smillie [MS] for realization of every admissible tuple by some pseudo-Anosov stable foliation. It refers to $L(\operatorname{Mod}_g)$ without defining whether that means dilatation or its logarithm, and then introduces a following unprovided question. No missing refinement statement is invented.

Short English statement

Is the least-dilatation pseudo-Anosov unique up to the stated conjugacy relation? The source's genus-one uniqueness claim needs an additional sign or equivalence convention. The following admissible-prong-data discussion is retained as a refinement program, without replacing the main uniqueness question.

Sources

• Local UnsolvedMath registry, permanent ID $11000080$, exact source code catalogue entry, Question 7.4, PDF page 57. No individual proposer or proposed year is supplied.
• 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, repeated local numbering distinguished by PDF page. The extract ends with chapter/page furniture and an introduction to another question.
• Internal reference Masur--Smillie [MS] for stable-foliation singularity-data realization; the full entry and admissibility hypotheses are absent.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied OPEN-TRIAGE summary says some small-genus cases may have uniqueness and general uniqueness is open, without exact cases or references.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.