Cat(0) polygonal complexes are 2-median

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 spac...

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Bibliographic Details
Published in:Geometriae dedicata 2024-02, Vol.218 (1), Article 5
Main Authors: Bader, Shaked, Lazarovich, Nir
Format: Article
Language:English
Subjects:
Algebraic Geometry
Buildings
Convex and Discrete Geometry
Differential Geometry
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Polygons
Projective Geometry
Topology
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Summary: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.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-023-00841-8