Unbounded Multiplicity in the Dilatation Spectrum of Every Term of the Johnson Filtration: An Observation on a Question of Farb
Year: 2026
Type: preprint
Abstract: Farb asked (Question 7.6 of "Some problems on mapping class groups and moduli space") whether spec(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 Mod_g, has bounded multiplicity for k ≥ 3; the multiplicity of a value is the number of conjugacy classes of Mod_g which realise it. We observe that the question is answered in the negative: for every g ≥ 2, k ≥ 1 and N there are N pseudo-Anosov classes in I_g(k) with one dilatation which are pairwise non-conjugate in 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 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 ≤ 2, written out with proofs, and explicit instances, which are computer-assisted, in exact integer arithmetic: for example, 64 classes in I_2(4) have one dilatation λ, where λ + 1/λ 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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-109-0082 (Farb, Some problems on mapping class groups and moduli space, Proc. Sympos. Pure Math. 74 (2006), Question 7.6).
Author: Ferudun, Alper
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221453904
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