Question 3.14 — (Rational growth for properties).

Definitions and mathematical statement

Question 3.14 asks: for which properties $P$ is the source's growth function $f_P$ rational? The preceding definition of $f_P$ is absent. A standard conditional interpretation fixes a finite symmetric generating set $A$ of $\operatorname{Mod}_g$, lets $|h|_A$ be word length, and forms \[ a_n(P)=\#\{h\in\operatorname{Mod}_g:|h|_A=n,\ h\text{ has }P\}, \qquad f_P(t)=\sum_{n\geq0}a_n(P)t^n. \] Rationality then means that this formal series is the expansion of an element of ${\mathbb{Q}}(t)$ regular at $0$. Cumulative element counts give the series $f_P(t)/(1-t)$ and hence the same rationality question. Counting words or conjugacy classes instead need not give the same series. The excerpt does not establish which interpretation, generating set, or class of properties is intended.

The following extracted density discussion is context. For $S\subset\operatorname{Mod}_g$, set $B_A(r)=\{h:|h|_A\leq r\}$. Where the limits exist, it proposes \[ d_A(S)=\lim_{r\to\infty}\frac{\#(B_A(r)\cap S)}{\#B_A(r)},\qquad d_{\log,A}(S)=\lim_{r\to\infty} \frac{\log\#(B_A(r)\cap S)}{\log\#B_A(r)}. \] The logarithmic expression additionally requires nonzero numerator counts and denominator different from $1$ eventually. Existence of either limit is not automatic. The source expects zero ordinary density for many sets because of exponential counts; this is motivation rather than a quantified theorem.

The source also claims generating-set independence of positivity for subgroup density. In its displayed intersection formula only the ambient generating set occurs, although the prose mentions a generating set for the subgroup as well; that discrepancy requires clarification. The passage ends by announcing a folklore conjecture whose statement is not included, and no such conjecture is supplied here.

Short English statement

Determine which mapping-class properties have rational growth series, once the missing counting convention is specified. The subsequent ordinary and logarithmic density definitions are retained with their limit-existence qualifications.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000035$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Question 3.14, PDF page 35: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Supplied access date: 2026-07-29; extraction: pdf-font-structure; page-disambiguated numbering among 329 headings.
• The earlier definition of $f_P$, referenced footnote 4 and the announced following conjecture are absent from the record. No additional primary citation is supplied; the registry says it could not verify the status without the statement.