Cat(0) polygonal complexes are 2-median

Shaked Bader, Nir Lazarovich

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Article number5
JournalGeometriae Dedicata
Volume218
Issue number1
DOIs
StatePublished - 24 Oct 2023

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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