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

Embeddings of cat(0) cube complexes in products of trees
Embeddings of cat(0) cube complexes in products of trees
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.
Holloway, Gemma Lauren
b8021059-be93-4a41-b13d-5003753f09b0
Holloway, Gemma Lauren
b8021059-be93-4a41-b13d-5003753f09b0
Niblo, Graham
43fe9561-c483-4cdf-bee5-0de388b78944

Holloway, Gemma Lauren (2007) Embeddings of cat(0) cube complexes in products of trees. University of Southampton, School of Mathematics, Doctoral Thesis, 107pp.

Record type: Thesis (Doctoral)

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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Published date: August 2007
Organisations: University of Southampton

Identifiers

Local EPrints ID: 66300
URI: https://eprints.soton.ac.uk/id/eprint/66300
PURE UUID: e43d7e6b-b2c5-45cc-b4bb-8664ddd0aa8e
ORCID for Graham Niblo: ORCID iD orcid.org/0000-0003-0648-7027

Catalogue record

Date deposited: 26 Mar 2010
Last modified: 13 Jun 2019 00:39

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