A new axiomatics for masures

TL;DR

This paper introduces a simplified axiomatic framework for masures, which are geometric structures generalizing Bruhat-Tits buildings, by exploring intersection properties of apartments in these spaces.

Contribution

It provides a new, simpler set of axioms for masures, extending known properties of apartment intersections in buildings to this broader context.

Findings

Proves intersection properties of apartments in certain masures

Deduces a new, simplified axiomatic system for masures

Extends concepts from Bruhat-Tits buildings to Kac-Moody group settings

Abstract

Masures are generalizations of Bruhat-Tits buildings. They were introduced to study Kac-Moody groups over ultrametric fields, which generalize reductive groups over the same fields. If A and A are two apartments in a building, their intersection is convex (as a subset of the finite dimensional affine space A) and there exists an isomorphism from A to A fixing this intersection. We study this question for masures and prove that the analogous statement is true in some particular cases. We deduce a new axiomatic of masures, simpler than the one given by Rousseau.