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 $\theta \in H^1(X,\mathbb{Z}\mathrm{/}2\mathbb{Z})$, 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 $Prym:{\mathcal{R}}_g\to {\mathcal{A}}_{g-1}$. For $g\ge 4$ and $h\le g-1$, we show that $Prym$ is the unique nonconstant holomorphic map of complex orbifolds $F:{\mathcal{R}}_g\to {\mathcal{A}}_h$. This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms ${\pi }_1^{orb}({\mathcal{R}}_g)\to Sp(2h,\mathbb{Z})$ for $h\le g-1$. This is achieved using arguments from geometric group theory and low-dimensional topology.