A proof of Sageev's Theorem on hyperplanes in CAT(0) cubical complexes

Details A proof of Sageev's Theorem on hyperplanes in CAT(0) cubical complexes A proof of Sageev's Theorem on hyperplanes in CAT(0) cubical complexes

2009-09-04

arxiv.org/abs/0909.0968v2

Mathematics

Geometric Topology

14 pages, 1 figure; updated abstract and introduction to credit Chepoi and Gerasimov for proving CAT(0) cubical complex = disc

Scientific paper

We prove that a hyperplane in a CAT(0) cubical complex X has no self-intersections and separates X into two convex complementary components. These facts were originally proved by Sageev. Our argument shows that his theorem is a corollary of Gromov's link condition. We also give new arguments establishing some combinatorial properties of hyperplanes. We show that these properties are sufficient to prove that the 0-skeleton of any CAT(0) cubical complex is a discrete median algebra, a fact that has previously been proved by Chepoi, Gerasimov, and Roller.

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