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Harmonic Splittings of Surfaces

Benson Farb·Michael Wolf
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Abstract

We give a proof, using harmonic maps from disks to real trees, of Skora's theorem (Morgan-Otal (1993), Skora (1990), originally conjectured by Shalen): if G is the fundamental group of a surface of genus at least 2, then any small minimal G-action on a real tree is dual to the lift of a measured foliation. Analytic tools like the maximum principle are used to simplify the usual combinatorial topology arguments. Other analytic objects associated to a harmonic map, such as the Hopf differential and the moduli space of harmonic maps, are also introduced as tools for understanding the action of surface groups on trees.

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Benson Farb, Michael Wolf. 2000-03-08. Harmonic Splittings of Surfaces. https://arxiv.org/abs/math/0003051

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