Holomorphic maps between configuration spaces of Riemann surfaces

Abstract 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 $$\operatorname {Conf}_n(\mathbb {C})\rightarrow \operatorname {Conf}_m(\mathbb {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>Conf</mml:mo> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:msub> <mml:mo>Conf</mml:mo> <mml:mi>m</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> provided that $$n\ge 5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> and $$m\le 2n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> extending the Tameness Theorem of Lin, which is the case $$m = n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . 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.