Constructing Buildings and Harmonic Maps

Abstract

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 $\varphi$-harmonic map from a Riemann surface, in the case of two-dimensional buildings for the group $S{L}_3$. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for $S{L}_2$. Our conjectural construction would determine the exponents for $S{L}_3$ WKB problems, and it can be put into practice on examples.

61 pages, 24 figures. Comments are welcome