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

TL;DR

This paper provides a new proof of Sageev's theorem on hyperplanes in CAT(0) cubical complexes using Gromov's link condition, and explores related combinatorial properties.

Contribution

It offers a novel proof of Sageev's theorem and demonstrates that hyperplanes' properties imply the median algebra structure of the complex's 0-skeleton.

Findings

Hyperplanes have no self-intersections and separate the complex into convex parts.

The 0-skeleton forms a discrete median algebra.

New combinatorial arguments support existing theorems.

Abstract

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.