Masures affines

TL;DR

This paper introduces an abstract framework for affine hovels, generalizing affine buildings, and explores their structural properties including retractions, twin buildings at infinity, and residue structures.

Contribution

It provides a unified abstract definition of affine hovels, extending previous concepts to include non-simplicial cases and Kac-Moody group constructions.

Findings

Existence of retractions with a sector germ center.

Addition of twin buildings or microaffine buildings at infinity.

Residue structures at points form pairs of twin buildings.

Abstract

We give an abstract definition of affine hovels which generalizes the definition of affine buildings (eventually non simplicial) given by Jacques Tits and includes the hovels built by Stephane Gaussent and the author for some Kac-Moody groups over ultrametric fields. We prove that, in such an affine hovel I, there exist retractions with center a sector germ and that we can add at the infinity of I a pair of twin buildings or two microaffine buildings. For some affine hovels I, we prove that the residue at a point of I has a natural structure of pair of twin buildings and that there exists on I a preorder which induces on each apartment the preorder associated to the Tits cone.