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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.

30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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