CAT(0) Polygonal Complexes are 2-Median

TL;DR

This paper introduces the concept of 2-median spaces and proves that CAT(0) Euclidean polygonal complexes, including rank-2 affine buildings, satisfy this property, extending median space theory.

Contribution

It defines 2-median spaces and demonstrates that CAT(0) polygonal complexes are 2-median, broadening understanding of median-like properties in non-positive curvature spaces.

Findings

CAT(0) Euclidean polygonal complexes are 2-median.

Rank-2 affine buildings are 2-median.

Elementary proof of minimal disc injectivity.

Abstract

Median spaces are spaces in which for every three points the three intervals between them intersect at a single point. It is well known that rank-1 affine buildings are median spaces, but by a result of Haettel, higher rank buildings are not even coarse median. We define the notion of ``2-median space'', which roughly says that for every four points the minimal discs filling the four geodesic triangles they span intersect in a point or a geodesic segment. We show that CAT(0) Euclidean polygonal complexes, and in particular rank-2 affine buildings, are 2-median. In the appendix, we recover a special case of a result of Stadler of a Fary-Milnor type theorem and show in elementary tools that a minimal disc filling a geodesic triangle is injective.