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Higher laminations and affine buildings. (English) Zbl 1348.30023
Summary: We give a Thurston-like definition for laminations on higher Teichmüller spaces associated to a surface $$S$$ and a semi-simple group $$G$$ for $$G=\operatorname{SL}_m$$ or $$\operatorname{PGL}_m$$. The case $$G=\operatorname{SL}_2$$ or $$\operatorname{PGL}_2$$ corresponds to the classical theory of laminations on a hyperbolic surface. Our construction involves positive configurations of points in the affine building. We show that these laminations are parametrized by the tropical points of the spaces $$\mathcal{X}_{G,S}$$ and $$\mathcal{A}_{G,S}$$ of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
##### Keywords:
higher Teichmüller theory; tropical points; laminations
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