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 $\operatorname{Conf}_n(\mathbb{C})\to \operatorname{Conf}_m(\mathbb{C})$ provided that $n\ge 5$ and $m\le 2n$ extending the Tameness Theorem of Lin, which is the case $m = n$. 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.