Holomorphic maps between configuration spaces of Riemann surfaces

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps Confₙ(C)→ Confₘ(C) Conf n ( C ) → Conf m ( C ) provided that n≥ 5 n ≥ 5 and m≤ 2n m ≤ 2 n extending the Tameness Theorem of Lin, which is the case $$m = n$$ m = n . We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious “effective de Franchis problem”. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.