Abstract This paper studies the relationship between an analytic compactification of the moduli space of flat SL₂ (C) SL 2 (C) connections on a closed, oriented 3-manifold M M defined by Taubes, and the Morgan–Shalen compactification of the SL₂ (C) SL 2 (C) character variety of the fundamental group of M M. We exhibit an explicit correspondence between Z/2 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 Z/2 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 Z/2 Z / 2 harmonic 1-forms but which have compact SL₂ (C) SL 2 (C) character varieties, and this resolves a folklore conjecture.
He et al. (Fri,) studied this question.
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