Unbounded Multiplicity in the Dilatation Spectrum of Every Term of the Johnson Filtration: An Observation on a Question of Farb

Farb asked (Question 7.6 of Some problems on mapping class groups and moduli space) whether \(\operatorname{spec}(\mathcal{I}_g(k))\), the set of the logarithms of the dilatations of the pseudo-Anosov elements of the \(k\)-th term of the Johnson filtration of the mapping class group \(\operatorname{Mod}_g\), has bounded multiplicity for \(k\ge3\); the multiplicity of a value is the number of conjugacy classes of \(\operatorname{Mod}_g\) which realise it. We observe that the question is answered in the negative: for every \(g\ge2\), \(k\ge1\) and \(N\) there are \(N\) pseudo-Anosov classes in \(\mathcal{I}_g(k)\) with one dilatation which are pairwise non-conjugate in \(\operatorname{Mod}_g\), also up to inversion and in the extended mapping class group. This is an observation: the negative answer follows from the mechanism which the source itself gives for the Torelli group and the Johnson kernel (Thurston's representation of the group generated by the twists about two filling curves, equal traces of non-conjugate elements, the two curves chosen separating), applied to a two-generator subgroup of a deep term of the lower central series of that group, which the source itself places in \(\mathcal{I}_g(k)\), with a reference to Farb, Leininger and Margalit. What the note adds is the comparison of conjugacy in the Fuchsian subgroup with conjugacy in the mapping class group, which the source leaves implicit also for \(k\le2\), written out with proofs, and explicit instances, which are computer-assisted, in exact integer arithmetic: for example, \(64\) classes in \(\mathcal{I}_2(4)\) have one dilatation \(\lambda\), where \(\lambda+\lambda^{-1}\) is an integer with \(5116\) digits. The answer is apparently not recorded in the literature accessible to us, and it may well be known to specialists. Question 7.7 of the source, on the simple length spectrum, is not answered. This is an unrefereed note.