Finiteness and integrality of fixed points under finite-index subgroups

Establish that for g≥3 and any finite-index subgroup Γ⊂Mod_{g,n}, the action of Γ on Y(g,n,r)^{irr} has only finitely many fixed points, and that all such fixed points correspond to local systems defined over the ring of integers \mathscr{O}_K of some number field K.

Background

This conjecture gives a quantitative form of non-abelian big monodromy, asserting finiteness of fixed points under large subgroups, along with an arithmetic integrality property for the corresponding local systems, mirroring results known for rigid local systems.

References

Conjecture Let $g\geq 3$, and $\Gamma\subset\on{Mod}_{g,n}$ a finite index subgroup. Then $\Gamma$ acts on $Y(g,n,r){\text{irr}}$ with only finitely many fixed points. All such fixed points correspond to local systems defined over the ring of integers of some number field $\mathscr{O}_K$.

— Motives, mapping class groups, and monodromy (2409.02234 - Litt, 3 Sep 2024) in Conjecture, Section 6.1