Question 7.6 — Does spec(Ig(k)) have bounded multiplicity for k≥ 3?

Definitions and mathematical statement

Question 7.6 asks whether $\operatorname{spec}(\mathcal I_g(k))$ has bounded multiplicity for $k\geq3$. Neither the subgroup notation $\mathcal I_g(k)$ nor the spectrum and its counting convention are defined in the supplied excerpt.

In the natural Teichmüller-length interpretation, a pseudo-Anosov $\varphi$ with stretch factor $\lambda(\varphi)>1$ has length $\ell(\varphi)=\log\lambda(\varphi)$. After specifying which conjugacy classes or closed geodesics are counted, write $m_{g,k}(t)$ for the number of those classes from $\mathcal I_g(k)$ with length $t$. Bounded multiplicity for each fixed $k$ would mean \[ \forall k\geq3\quad\exists C_{g,k}<\infty\quad \forall t>0,\qquad m_{g,k}(t)\leq C_{g,k}. \] The source does not prescribe a bound uniform in $g$ or $k$. This formulation is conditional on the missing spectrum definition. Counting conjugacy inside $\mathcal I_g(k)$ rather than inside $\operatorname{Mod}_g$, distinguishing inverse classes, and counting primitive classes rather than all powers can change multiplicity. A set of length values alone does not retain multiplicity.

The source suggests considering instead the simple length spectrum: the part of $\operatorname{spec}(\operatorname{Mod}_g)$ arising from pseudo-Anosovs represented by non-self-intersecting geodesic loops in $\mathcal M_g$. Simplicity here concerns loops in moduli space, not simple closed curves on the underlying surface. The excerpt does not state a bounded-multiplicity theorem for this simple spectrum or define the precise geodesic/orbifold convention.

Short English statement

For each filtration level $k\geq3$, is there a uniform bound on how many counted pseudo-Anosov classes have any one spectral length? The missing subgroup and multiplicity conventions are essential.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $11000082$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Question 7.6, PDF page 58: 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.
• No additional reference is supplied. The registry explicitly says it cannot verify the exact invariant; preceding definitions are absent.