Andersen–Masbaum–Ueno (AMU) Conjecture for Mapping Class Groups

Establish that for any compact oriented surface Σ_{g,n} and any mapping class f in Mod(Σ_{g,n}), f has a pseudo-Anosov part if and only if the SO(3)-Witten–Reshetikhin–Turaev quantum representation ρ_r(f) has infinite order for all sufficiently large odd integers r ≥ 3.

Background

The paper studies the SO(3)-Witten–Reshetikhin–Turaev (WRT) quantum representations ρr of mapping class groups Mod(Σ{g,n}). The Nielsen–Thurston classification partitions mapping classes into finite order, reducible, and pseudo-Anosov types. The AMU conjecture posits a spectral criterion for detecting the pseudo-Anosov part via the asymptotic behavior (as r grows) of these quantum representations.

The authors prove this conjecture for prime r and for all nontrivial f in the derived subgroup [J_2(Σ{g,n}), J_2(Σ{g,n})] of the Johnson kernel, yielding new cases (including closed surfaces). Prior progress confirmed the conjecture in specific low-complexity settings (e.g., Σ{0,4}, Σ{1,1}, and certain Σ_{0,n}). The general AMU conjecture, however, remains open beyond these cases and the new family proven in this work.

References

The Andersen-Masbaum-Ueno conjecture (or AMU conjecture) is then the following: Let Σ{g,n} be a compact oriented surface and f∈ Mod(Σ{g,n}). Then f has a pseudo-Anosov part if and only if ρ_r(f) has infinite order for all large enough odd r≥3.

— The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel  (2603.29397 - Detcherry, 31 Mar 2026) in Section 1 (Introduction), Conjecture (AMU conjecture)

The AMU conjecture may be stated as follows: Let $f\in \mathrm{Mod}(S)$ be a mapping class with a pseudo-Anosov part. Then $\rho_p(f)$ has infinite order for all primes $p>>1$.

— Ivanov's finite-index Frattini subgroup conjecture for subgroups of the Torelli group  (2610.01734 - Detcherry, 1 Oct 2026) in Section 1, Introduction; Conjecture 1.2