Constructing Buildings and Harmonic Maps

In a continuation of our previous work, we outline a theory which should lead to the construction of a universal pre-building and versal building with a $ϕ$-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group $SL_3$. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for $SL_2$. Our conjectural construction would determine the exponents for $SL_3$ WKB problems, and it can be put into practice on examples.

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