Embeddings of cat(0) cube complexes in products of trees

Abstract

In ‘Groups acting on connected cubes and Kazhdan’s property T’, [29], Niblo and Roller showed that any CAT(0) cube complex embeds combinatorially and quasi-isometrically in the Hilbert space '2(H) where H is the set of hyperplanes. This Hilbert space may be viewed as the completion of an infinite product of trees. In this thesis, we consider the question of the existence of quasi-isometric maps from CAT(0) cube complexes to finite products of trees, restricting our attention to folding maps as used in [29].

Following an overview of the properties of CAT(0) cube complexes, we first prove that there exists CAT(0) square complexes which do not fold into a product of trees with fewer than k factors for arbitrary k, giving examples which admit co-compact proper actions by right-angled Coxeter groups. We also show that there exists a CAT(0) square complex which does not fold into any finite product of trees.

We then identify a class of group actions on CAT(0) cube complexes for which the existence of such an action implies the existence of a quasi-isometric embedding of that group in a finite product of finitely branching trees. We apply this result to surface groups, certain 3-manifold groups and more generally to Coxeter groups which do not contain affine triangle subgroups.

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Identifiers

ORCID for Graham Niblo: orcid.org/0000-0003-0648-7027

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