Geometry of Positive Configurations in Affine Buildings

TL;DR

This paper explores the geometry of positive point configurations in affine buildings, linking canonical functions from algebraic geometry to metric structures, and proposes conjectures connecting higher laminations with affine building geometry.

Contribution

It provides a self-contained definition of positive configurations in affine buildings, studies their geometry, and formulates conjectures relating tropicalized canonical functions to affine building metrics.

Findings

Proved some conjectures relating tropical functions to affine building geometry.

Connected algebraic valuations of lattices with geometric structures in affine buildings.

Identified the role of minimal networks and max-flow/min-cut analogies in the conjectures.

Abstract

Positive configurations of points in the affine building were introduced in \cite{Le} as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of these configurations. The canonical functions on triples of flags that were defined by Fock and Goncharov in \cite{FG1} have a tropicalization that gives functions on triples of points in the affine Grassmannian. One expects that these functions, though of algebro-geometric origin, have a simple description in terms of the metric structure on the corresponding affine building. We give a several conjectures describing the tropicalized canonical functions in terms of the geometry of affine buildings, and give proofs of some of them. The statements involve minimal networks and have some resemblance to the max-flow/min-cut theorem, which also plays a role in the proofs in unexpected ways. The conjectures can be reduced to purely algebraic statements about valuations of lattices that we argue are interesting in their own right. One can view these conjectures as the first examples of intersection pairings between higher laminations. They fit within the framework of the Duality Conjectures of \cite{FG1}.