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arXiv · 2212.00894

CAT(0) Polygonal Complexes are 2-Median

Shaked Bader·Nir Lazarovich
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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.

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Shaked Bader, Nir Lazarovich. 2022-12-01. CAT(0) Polygonal Complexes are 2-Median. https://doi.org/10.1007/s10711-023-00841-8

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