The geometry of cube complexes and the complexity of their fundamental groups
type: Item
instance of: scholarly article
title: The geometry of cube complexes and the complexity of their fundamental groups
zbMATH Open document ID: 0911.57002
DOI: 10.1016/S0040-9383(97)00018-9
published in: Topology
publication date: 1998-07-12T00:00:00Z
review text: The main result of this paper is that if \(G\) is a group acting effectively, cellularly, properly discontinuously, and cocompactly on a simply connected, nonpositively curved cube complex, then there is a synchronously biautomatic structure on \(G\). This unifies and generalizes a number of cases which had previously been shown to be automatic or biautomatic, such as free groups, free abelian groups, right angled Coxeter groups, right angled Artin groups, word hyperbolic Coxeter groups, \(T(4)\)-\(C(4)\)-\(P\) small cancellation groups, and many 3-manifold groups. The proof of the main result involves the development of some geometric properties of hyperplanes in simply-connected nonpositively curved cube complexes, and the definition of a ``normal cube-path'' connecting two vertices. The latter is a path that moves as diagonally as possible through the cube structure. The authors show that normal cube-paths satisfy the \(k\)-fellow traveler property, in fact that they satisfy it for \(k=1\).
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E99
zbMATH DE Number: 1140357
zbMATH Keywords: cube complex
zbMATH Keywords: CAT(0)
zbMATH Keywords: automatic
zbMATH Keywords: biautomatic
zbMATH Keywords: hyperbolic
zbMATH Keywords: synchronous
zbMATH Keywords: nonpositively curved
zbMATH Keywords: negatively curved
zbMATH Keywords: hyperplane
zbMATH Keywords: cube-path
author: Graham Niblo
author: Lawrence Reeves
reviewed by: Darryl McCullough
MaRDI profile type: Publication
full work available at URL: s0040-9383(97)00018-9
OpenAlex ID: W2053265348
description: scientific article; zbMATH DE number 1140357