The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel

Abstract: A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $Σ{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(Σ{g,n}),$ the matrix $ρr(f)$ has infinite order for any large $r,$ where $ρ_r$ is the $\mathrm{SO}(3)$-WRT quantum representation of the mapping class group $\mathrm{Mod}(Σ{g,n})$ at a primitive $r$-th root of unity. We prove this conjecture for prime $r$ and any $f\in [J_2(Σ{g,n}),J_2(Σ{g,n})],$ where $J_2(Σ_{g,n})$ is the Johnson kernel.

• The paper establishes that for any nontrivial element in the derived subgroup of the Johnson kernel, the SO(3)-WRT quantum representation has infinite order for large prime moduli.
• It employs an arithmetic approach with cyclotomic congruences and norm computations to link pseudo-Anosov mapping classes with their quantum images.
• The results extend the AMU conjecture to higher-genus closed surfaces, suggesting new pathways for exploring deeper levels of the Johnson filtration.

The Andersen-Masbaum-Ueno Conjecture for the Derived Subgroup of the Johnson Kernel

Introduction and Problem Setting

The paper investigates the Andersen-Masbaum-Ueno (AMU) conjecture in the context of the mapping class group $\mathrm{Mod}(\Sigma_{g,n})$ of a compact oriented surface $\Sigma_{g,n}$. The AMU conjecture stipulates that for a pseudo-Anosov mapping class $f$, the $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev (WRT) quantum representation $\rho_r(f)$ has infinite order for all sufficiently large odd integers $r$. This conjecture connects geometric properties (pseudo-Anosov dynamics) to quantum representations, specifically asking whether such topological actions can be characterized by the growth of quantum representations.

Historically, the AMU conjecture has been resolved in relatively few nontrivial cases, typically low-genus or punctured surface settings, and often by relating quantum representations to homological representations or by asymptotic analysis of representation traces. Previous positive resolutions relied on the homological content and explicit computation or geometric interpretation of mapping classes.

This work significantly extends the known positive results by verifying the AMU conjecture for the derived subgroup of the Johnson kernel $[J_2(\Sigma_{g,n}), J_2(\Sigma_{g,n})]$ and, as an immediate corollary, for any mapping class with a pseudo-Anosov part acting trivially mod $[J_2(\Sigma_{g,n}), J_2(\Sigma_{g,n})]$.

Main Results

The central theorem establishes that for any compact oriented surface $\Sigma_{g,n}$, any nontrivial $f$ in $\Sigma_{g,n}$0, and all sufficiently large prime $\Sigma_{g,n}$1, the matrix $\Sigma_{g,n}$2 under the $\Sigma_{g,n}$3-WRT quantum representation has infinite order. This statement is formalized as follows:

Theorem: Let $\Sigma_{g,n}$4 be non-trivial. Then for all sufficiently large prime $\Sigma_{g,n}$5, $\Sigma_{g,n}$6 has infinite order.

This result provides, for the first time, a verification of the AMU conjecture for a natural infinite index subgroup of $\Sigma_{g,n}$7 which is not captured by earlier methods.

Additional results include:

• Corollary: If $\Sigma_{g,n}$8 has a pseudo-Anosov part and finite order in the quotient group $\Sigma_{g,n}$9, then for large primes $f$0, $f$1 has infinite order.
• Theorem: For nontrivial $f$2, for all large enough primes $f$3, either $f$4 has infinite order or order exactly $f$5. This is optimal because separating Dehn twists map to elements of order $f$6.

Methods and Proof Strategy

The paper departs from previous analytic and geometric strategies for the AMU conjecture and utilizes techniques centered on number theory and the arithmetic of cyclotomic rings. The author leverages these key tools:

• Johnson Filtration and Johnson Kernel: $f$7 denotes the Johnson kernel, the subgroup generated by Dehn twists along separating curves. The derived subgroup $f$8 is the second commutator subgroup.
• Properties of $f$9 on $\mathrm{SO}(3)$0: Prior work had shown that, modulo $\mathrm{SO}(3)$1, $\mathrm{SO}(3)$2 acts trivially on $\mathrm{SO}(3)$3, but this congruence is refined at the level of the derived subgroup.
• A Novel Number-Theoretic Criterion: The core technical device is a criterion for a matrix $\mathrm{SO}(3)$4 to have infinite order if it is congruent to the identity mod $\mathrm{SO}(3)$5 but not equal to the identity. The argument uses explicit norm computations in cyclotomic fields and a careful analysis of divisibility properties.

Central to the proof is the observation that any $\mathrm{SO}(3)$6 acts (via quantum representation) as identity modulo $\mathrm{SO}(3)$7 but is nontrivial for large $\mathrm{SO}(3)$8. The asymptotic faithfulness of WRT representations ensures that for large prime $\mathrm{SO}(3)$9, such $\rho_r(f)$0 are nontrivial in all but the center. The number-theoretic criterion guarantees that such elements yield representations of infinite order.

Implications and Connections

This work offers several theoretically significant advances:

• First General AMU Verification in Higher Genus and Closed Surfaces: Earlier positive results for the AMU conjecture either relied on low genus, boundary components, or on explicit monodromy constructions of fibered links. This work delivers the first result for a family of subgroups naturally defined in terms of the mapping class group alone, and, notably, for closed surfaces.
• Extension to the Johnson Filtration: The techniques here, especially the interplay between algebraic congruence conditions and representation theory, suggest possible extensions to deeper terms of the Johnson filtration (e.g., $\rho_r(f)$1), contingent on further arithmetic analysis and properties of the Torelli group.
• Moduli of Torsion Pseudo-Anosov Classes: The corollaries give rise to explicit constructions of elements with pseudo-Anosov dynamics whose quantum images have prescribed order growth, enriching the known zoo of such elements in mapping class groups.

On a practical level, this tight link between quantum representations and the lower central series of the mapping class group may have ramifications for quantum invariants of three-manifolds, as well as for computational problems involving mapping class group representations in TQFT.

Future Directions

The arithmetic criterion for infinite order in cyclotomic unitary representations could be further exploited to analyze other natural subgroups of the mapping class group (such as higher Johnson kernels or Torelli subgroups). Moreover, the methods may shed light on questions relating the quantum representation image to the structure of the group itself, potentially uncovering new invariants or constraints.

Establishing the conjecture at even deeper levels of the Johnson filtration would refine understanding of the structure of mapping class groups as seen through the lens of quantum topology.

Conclusion

The paper establishes the AMU conjecture for the derived subgroup of the Johnson kernel by deploying an arithmetic argument involving congruences in cyclotomic rings and the asymptotic faithfulness of quantum representations. This sharpens the link between mapping class group structure and quantum representation theory, expanding the scope of known cases where the AMU conjecture holds, and highlighting new algebraic and number-theoretic phenomena at the interface of surface topology and quantum invariants (2603.29397).