I’ll discuss ongoing joint work with Aaron Landesman on the Putman-Wieland conjecture. Let f: S’–>S be a (possibly ramified) surjective map of smooth compact orientable surfaces, ramified along a finite set D, where S has genus g>1. Then the Putman-Wieland conjecture predicts that the (virtual) action of the mapping class group of (S,D) on the homology of S’ has no non-zero vectors with finite orbit. The conjecture is known to be false for g=2 due to recent work of Markovic. I’ll discuss some positive results in the case where the degree of f is small relative to the genus g. More generally, if f is Galois with deck transformation group G, I’ll explain how to verify the conjecture for the subspace of H_1(S’) spanned by G-representations of dimension at most g. The proof is Hodge-theoretic in nature.