Problem 4.1 — Show that there are many quasihomomorphisms f:MCG (Sg)→ R that satisfy (1) and (2) above plus (3) f is bounded on eve…

Definitions and mathematical statement

Let $G=\operatorname{MCG}(S_g)$ in the source's surface convention. A real quasihomomorphism is a function $f:G\to{\mathbb{R}}$ with finite defect \[D(f)=\sup_{a,b\in G}|f(ab)-f(a)-f(b)|<\infty.\] Homogeneity $f(a^n)=nf(a)$ is an additional condition and is not imposed unless it is part of the missing source assumptions.

The task is to show that there are "many" such functions satisfying the preceding conditions (1) and (2), together with \[\forall q,\quad\sup_{h\in G(q)}|f(h)|<\infty.\tag{3}\] Here $G(q)$ and the allowed $q$ are defined only in omitted preceding text. The displayed boundedness permits a constant depending on $q$ and $f$; a single constant uniform in $q$ is not stated. If $P_1,P_2$ denote the unreproduced conditions, the requested family lies in \[\{f:G\to{\mathbb{R}}:D(f)<\infty,\ P_1(f),\ P_2(f),\ \forall q\ \exists C_{f,q}<\infty\ \forall h\in G(q),\ |f(h)|\leq C_{f,q}\}.\] The symbols $P_1,P_2$ are explicit placeholders, not definitions of the missing properties. "Many" could refer to a vector-space dimension or another quantitative conclusion, but the excerpt supplies no criterion. It cannot silently be replaced by mere existence, unboundedness, homogeneous independence or an infinite-dimensional quotient.

The author remarks that a consequence about product sets of the $G(q)$'s was proved by a different method by Bestvina--Feighn, unpublished. The consequence itself is not supplied, so the reported result does not resolve the full quasihomomorphism construction as stated here. The summary's Bestvina--Fujiwara bounded-cohomology context is separate from that unpublished attribution.

Short English statement

Construct many quasihomomorphisms satisfying two missing conditions and boundedness on each specified set $G(q)$. Neither those sets nor the intended size of the family is defined in the truncated record.

Sources

• Local UnsolvedMath registry, permanent ID $11000007$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Problem 4.1, PDF page 15: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• Bestvina--Feighn, unpublished, is the source's attribution for an omitted product-set consequence. No bibliographic reference is supplied.
• The research summary mentions Bestvina--Fujiwara quasihomomorphism/bounded-cohomology work and an extended Nielsen--Thurston context without exact citations or a match to the truncated statement.