Putman–Wieland conjecture on fixed vectors in branched covers

Prove that for g≥3 and n≥0, if \Sigma_{g'}\to\Sigma_g is a finite branched cover and Γ is the stabilizer (a finite-index subgroup of Mod_{g,n+1}) acting on H_1(\Sigma_{g'},\mathbb{Z}), then no nonzero vector in H_1(\Sigma_{g'},\mathbb{Z}) has finite orbit under Γ.

Background

This conjecture is equivalent to strong vanishing statements for invariants of certain monodromy representations and is closely tied to big monodromy phenomena for families of curves and the structure of mapping class groups. The paper outlines algebro-geometric reformulations and partial results.

References

Conjecture [{\u007f[Conjecture 1.2]{putman2013abelian}] Suppose $g\geq 3$ and $n\geq 0$. Then there is no non-zero vector in $H_1(\Sigma_{g'}, \mathbb{Z})$ with finite $\Gamma$-orbit, under the action of $\Gamma$ on $H_1(\Sigma_{g'},\mathbb{Z})$ described above.

— Motives, mapping class groups, and monodromy (2409.02234 - Litt, 3 Sep 2024) in Conjecture (Putman–Wieland), Section 6.2