Constructing buildings and harmonic maps

Abstract

In a continuation of our previous work [21], 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 twodimensional buildings for the group SL3. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for SL2. Our conjectural construction would determine the exponents for SL3 WKB problems, and it can be put into practice on examples.