On the uniqueness of the Prym map

The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $\theta \in H^1(X, \mathbb{Z} / 2 \mathbb{Z})$ , a principally polarized abelian variety (PPAV) Prym $(X, \theta)$ . Denote the moduli space of pairs $(X, \theta)$ by $\mathcal{R}_g$ , and let $\mathcal{A}_h$ be the moduli space of PPAVs of dimension $h$ . The Prym construction globalizes to a holomorphic map of complex orbifolds Prym: $\mathcal{R}_g \rightarrow \mathcal{A}_{g-1}$ . For $g \geq 4$ and $h \leq g-1$ , we show that Prym is the unique nonconstant holomorphic map of complex orbifolds $F$ : $\mathcal{R}_g \rightarrow \mathcal{A}_h$ . This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms $\pi_1^{\text {arb }}\left(\mathcal{R}_g\right) \rightarrow \mathrm{Sp}(2 h, \mathbb{Z})$ for $h \leq g-1$ . This is achieved using arguments from geometric group theory and low-dimensional topology.