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

Abstract This paper studies the relationship between an analytic compactification of the moduli space of flat $\mathrm{SL}_{2}(\mathbb{C})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> connections on a closed, oriented 3-manifold $M$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> defined by Taubes, and the Morgan–Shalen compactification of the $\mathrm{SL}_{2}(\mathbb{C})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> character variety of the fundamental group of $M$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> . We exhibit an explicit correspondence between $\mathbb{Z}/2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> harmonic 1-forms, measured foliations, and equivariant harmonic maps to ℝ-trees, as initially proposed by Taubes. As an application, we prove that $\mathbb{Z}/2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> 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 $\mathbb{Z}/2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> harmonic 1-forms but which have compact $\mathrm{SL}_{2}(\mathbb{C})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:math> character varieties, and this resolves a folklore conjecture.