Constructing Buildings and Harmonic Maps

TL;DR

This paper proposes a theoretical framework for constructing universal and versal buildings with harmonic maps from Riemann surfaces for the group SL_3, generalizing classical foliation theories and aiding in WKB problem analysis.

Contribution

It introduces a new theory for constructing universal pre-buildings and harmonic maps for SL_3, extending classical foliation concepts to higher-dimensional buildings.

Findings

Framework outlined for SL_3 harmonic maps

Potential to determine exponents in SL_3 WKB problems

Practical examples demonstrate the theory's application

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 $\phi$-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.