$\mathbb{Z}/2$ harmonic 1-forms, ℝ-trees, and the Morgan-Shalen compactification

This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_{2}(\mathbb{C})$ connections on a closed, oriented 3-manifold $M$ defined by Taubes, and the Morgan–Shalen compactification of the $\mathrm{SL}_{2}(\mathbb{C})$ character variety of the fundamental group of $M$ . We exhibit an explicit correspondence between $\mathbb{Z}/2$ harmonic 1-forms, measured foliations, and equivariant harmonic maps to ℝ-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ harmonic 1-forms exist on all reducible or Haken manifolds with respect to all Riemannian metrics. We also prove the existence of manifolds which support singular $\mathbb{Z}/2$ harmonic 1-forms but which have compact $\mathrm{SL}_{2}(\mathbb{C})$ character varieties, and this resolves a folklore conjecture.