Abstract
We prove that any hyperplane H 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 was previously proved by Chepoi, Gerasimov, and Roller.
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Farley, D. (2016). A Proof of Sageev’s Theorem on Hyperplanes in CAT(0) Cubical Complexes. In: Davis, M., Fowler, J., Lafont, JF., Leary, I. (eds) Topology and Geometric Group Theory. Springer Proceedings in Mathematics & Statistics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-43674-6_4
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DOI: https://doi.org/10.1007/978-3-319-43674-6_4
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