Problem 2.11 — Let g≥ 3 and b≥ 1.

Definitions and mathematical statement

Let $S$ be an oriented genus-$g$ surface with $g\geq3$ and $b\geq1$ boundary components. In the standard relative-boundary convention, $\operatorname{Mod}_g^b$ consists of orientation-preserving mapping classes fixing the boundary pointwise. Let $a_1,a_2,\ldots$ be any infinite sequence of nonseparating simple closed curves, let $t_a$ denote the right Dehn twist, and put $P_n=t_{a_1}t_{a_2}\cdots t_{a_n}$. Curves may repeat. The function $c$ occurring in the source is not defined in this record, so the following statements remain conditional on its preceding definition.

Problem 2.11 asks whether, for every such sequence, \[ \lim_{n\to\infty}\frac{c(P_n)}n \] exists and, if it exists, is always positive. It further asks for a positive lower bound $c(P_n)\geq nK$ for every $n$. The subsequent discussion singles out the stronger uniform question \[ \exists K=K(g,b)>0\quad\forall(a_i)_{i\geq1}\quad \forall n\geq1,\qquad c(P_n)\geq nK. \] The earlier lower-bound sentence does not explicitly say whether its $K$ may depend on the sequence. Existence of a positive limit alone also does not supply a stated all-$n$ bound without further information on $c$.

The source observes that such a sequence-independent $K$ would bound the length of any displayed positive nonseparating-twist factorization $f=t_{a_1}\cdots t_{a_k}$ by $k\leq c(f)/K$, and says this would answer the omitted Problem 2.9 positively. It emphasizes that $b\geq1$ is essential: on a closed surface a positive identity factorization can be repeated to give the reported zero limit, using the properties of $c$ implicit in the source. Conversely it reports no nonempty positive identity factorization for the boundary-fixed group, citing an embedding in a larger closed surface and Smith [41], whose theorem is described as saying that curves in a positive identity factorization fill the closed surface. The source credits Rostislav Matveyev, June 2000, for communicating that fact.

Short English statement

For arbitrary products of successive positive nonseparating Dehn twists on a surface with boundary, does the source's complexity divided by word length have a positive limit, and is there a uniform positive linear lower bound? The definition of the complexity is missing.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000116$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 2.11, PDF page 97: 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.
• Supplied citation: Smith [41], with the filling statement credited as learned from Rostislav Matveyev in June 2000. Problem 2.9 and the definition of $c$ are absent, as is the full bibliography entry. The registry literature summary is truncated.