Article Dans Une Revue Journal of Combinatorial Theory, Series B Année : 2013

On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

Résumé

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Delta can be coloured with at most epsilon(Delta) = M Delta(26) colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most epsilon(Delta) trees, and that the event structure whose domain is X admits a nice labelling with epsilon(Delta) labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper

Dates et versions

hal-01199910 , version 1 (16-09-2015)

Identifiants

Citer

Victor Chepoi, Hagen Mark. On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes. Journal of Combinatorial Theory, Series B, 2013, 103 (4), pp.428-467. ⟨10.1016/j.jctb.2013.04.003⟩. ⟨hal-01199910⟩
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