Large-n big monodromy for higher Prym representations

Prove the conjectured large-n monodromy density: Establish the existence of a function c(g, dim ρ) such that, for a fixed genus g and an irreducible complex representation ρ: H → GL_r(ℂ) of a finite group H, if the number Δ of punctures of the base surface Σ_g with nontrivial local monodromy for the composition π_1(Σ_{g,n}) → H → GL_r(ℂ) satisfies Δ > c(g, dim ρ), then the image of the stabilizer Mod_φ of the surjection φ: π_1(Σ_{g,n}, x) ⇀ H inside GL(W^1H^1(Σ_{g,n}, ρ)) is Zariski-dense in the following group determined by the self-duality type of ρ: (i) SO(W^1H^1(Σ_{g,n}, ρ)) if ρ is symplectically self-dual; (ii) Sp(W^1H^1(Σ_{g,n}, ρ)) if ρ is orthogonally self-dual; or (iii) the product of SL(W^1H^1(Σ_{g,n}, ρ)) with a finite group of scalars if ρ is not self-dual.

Background

The paper studies the monodromy of mapping class group actions on H^1 of covering curves arising from finite Galois H-covers, generalizing Prym representations. For large base genus g relative to the maximal dimension of irreducible H-representations, the authors prove Zariski-density of the connected monodromy group, but they also propose an alternative regime where g is fixed and the number of punctures n grows.

Conjecture 1.4 formulates a precise large-n criterion in terms of the number Δ of punctures with nontrivial local monodromy under ρ∘φ, predicting Zariski-density of the stabilizer’s image in the appropriate classical group based on the self-duality type of ρ. The authors motivate this by connections to arithmetic statistics over function fields and note their theorem verifying nonexistence of finite-orbit vectors under a weaker bound on Δ.