Putman–Wieland conjecture on absence of finite-orbit vectors

Establish that for genus g > 2, the virtual action of the mapping class group Mod_{g,n+1} on H^1(Σ_{g'}, ℂ) associated to a finite Galois H-cover Σ_{g',n'} → Σ_{g,n} has no nonzero vectors with finite orbit.

Background

The authors situate their main "big monodromy" results within a broader philosophy and highlight the Putman–Wieland conjecture as a closely related open problem. The conjecture asserts that for g > 2, no nonzero vectors in the relevant cohomology have finite mapping class group orbits; it was originally stated also for g ≥ 2, but a counterexample exists for g = 2.

They note that their Zariski-density results imply instances of the conjecture for sufficiently large g relative to H, but the conjecture in full generality for g > 2 remains a significant open problem.