Counterexamples to Satoh's SL(2) representation-filtration conjecture

Abstract

Let A_n(k) be the integral Andreadakis filtration of the automorphism group of a free group, and let D_n(k) be the kernel of its action on J/J^{k+1} in the rational SL(2,ℂ) matrix-entry algebra at the trivial representation. We construct elements of D_n(5) \ A_n(5) for every n≥4, and of D_n(6) \ A_n(6) for every n≥3. Thus the equality in all degrees conjectured in Satoh’s 2018 report is false at every rank in its stated range. The construction uses a classical degree-five Lie identity, elementary free-group automorphisms, and integral Magnus coefficients. The proof works with actual powers of J, rather than only values along one-parameter representations.

The idea in plain English

A free group starts with symbols that can be combined into words, with no relations beyond cancellation. An automorphism changes the generators while preserving the group. Two filtrations measure how hard such a change is to detect: one uses successive group quotients, while the other uses two-by-two matrix representations near the trivial representation.

The paper gives explicit changes that the matrix test misses to a specified order even though the corresponding group quotient detects them. It treats degree five for every rank at least four, and degree six for every rank at least three.

The key idea is that a classical identity holds for two-by-two matrices but not for freely noncommuting symbols. Turning this mismatch into words gives counterexamples to the all-degree equality stated in Satoh’s 2018 report.

The precise result

For the free group F_n, let A_n(k) be the kernel of the action on F_n/Γ_{k+1}F_n, using the integral lower central series. Let D_n(k) be the kernel of the action on the entire J/J^{k+1}, where J is the augmentation ideal in the rational algebra of SL(2,ℂ) matrix-entry functions.

With group commutator [g,h]=ghg⁻¹h⁻¹, set w=[[[x₁,x₂],[x₁,x₃]],x₁]. For every n≥4, the automorphism x₄↦x₄w fixing the other generators lies in D_n(5) but not A_n(5).

Set v=[[[x₁,x₂],[x₁,[x₁,x₂]]],x₁]. For every n≥3, the automorphism x₃↦x₃v fixing the other generators lies in D_n(6) but not A_n(6). Thus equality in all degrees fails at every rank n≥3 covered by the stated conjecture.

Why the result holds

The classical identity [[[X,Y],[X,Z]],X]=0 for two-by-two matrices follows here directly from Cayley–Hamilton over any commutative coefficient ring. Substituting Z=[X,Y] gives the weighted degree-six identity.

A filtered group-commutator calculation transfers these identities to actual powers J⁶ and J⁷ in the matrix-entry algebra. It controls the whole ideal quotient, rather than only selected representations or one associated graded piece.

Integral Magnus expansions retain a nonzero degree-five coefficient for w and degree-six coefficient for v. These obstruct membership in the next lower central subgroup. Each word omits the generator that is moved, so the displayed substitutions have explicit inverses.

Scope and limitations

This is a result about the rational SL(2,ℂ) matrix-entry algebra and the full ideal quotient J/J^{k+1}. It is not a theorem about a trace algebra, a character algebra or a Grothendieck representation ring.

The theorem does not establish a degree-five counterexample at rank three, failure in every degree, or an extension to larger matrix sizes.

It addresses the representation-filtration equality formulated in 2018. It is distinct from the comparison of the Andreadakis filtration with the lower central series of IA_n.

The matrix identity and the general algebraic tools are classical. The stated contribution is their explicit application to this equality; no new polynomial identity, historical firstness, exhaustive novelty clearance or continuing worldwide openness is claimed.

Status and AI use

AI tools were used extensively in developing the constructions, checking proofs and computations, investigating sources and writing. Internal AI reviews do not constitute independent human peer review.

The manuscript and Zenodo publication-date metadata are dated 15 October 2026. The Zenodo record was created on the same date. These dates identify this edition; they do not establish first priority. Version 1.0 is the Zenodo version; the manuscript does not print a version number.

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I welcome substantive collaboration from researchers in free-group automorphisms and polynomial identities on independent verification, comparison with earlier literature and development toward journal review. Coauthorship would reflect actual research contributions and mutual agreement.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Counterexamples to Satoh's SL(2) representation-filtration conjecture (Zenodo version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23387442

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Background and prior work

• Satoh — On the Johnson homomorphisms of the automorphism groups of free groups: The 2018 Oberwolfach report, pp.87–90, supplies the component-function algebra, the full ideal quotient and the stated all-degree equality conjecture.
• Satoh — On the graded quotients of the SL(m,ℂ)-representation algebras of groups: The inspected author preprint, arXiv:1607.05411v2 dated 8 December 2017, supplies the precise filtration definitions and equality through degree four. It is identified separately from the related 2021 Pacific Journal of Mathematics publication; its final journal text was not inspected.
• Burde — Derivation double Lie algebras: Identity (4) in the inspected arXiv:1411.0950v2 author preprint, dated 27 February 2015, is the classical degree-five identity used here, after a sign and variable change. Theorem 4.14 attributes its place in an sl₂ identity basis to Razmyslov; this attribution is through Burde.