A simple proof of a theorem of Whyte.
type: Item
instance of: scholarly article
title: A simple proof of a theorem of Whyte.
zbMATH Open document ID: 1073.20033
author: Briana Cook
author: Eric Freden
author: Alisha McCann
published in: Geometriae Dedicata
publication date: 2005-03-15T00:00:00Z
review text: The authors give an elementary geometric proof of a theorem of \textit{K. Whyte} [Geom. Funct. Anal. 11, No. 6, 1327-1343 (2001; Zbl 1004.20024)] stating that all Baumslag-Solitar groups \(BS(p,q)\) with \(1<p<q\) are quasi-isometric. The authors' approach is to replace the Cayley graph of such a group with a \(2\)-complex \(X\) homeomorphic to a regular tree \(T\) cross the real line, where \(T\) is the Bass-Serre tree coming from the interpretation of \(BS(p,q)\) as an HNN extension of \(\mathbb{Z}\). Each \(2\)-complex is quasi-isometric to the group from which it arises. The authors show that basic moves such as collapses and slides on the edges of \(T\) can be lifted to \(X\) while preserving the quasi-isometry class. They finish the proof by lifting a standard slide argument showing that all regular trees of finite valence are quasi-isometric to \(X\). Thus, they conclude that \(BS(2,3)\) is quasi-isometric to \(BS(p,q)\) where \(1<p<q\) and that all \(BS(n,n)\) with \(n>1\) belong to a single isometry class, distinct from \(BS(2,3).\)
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 2145816
zbMATH Keywords: Baumslag-Solitar groups
zbMATH Keywords: quasi-isometry classes
zbMATH Keywords: graphs of groups
zbMATH Keywords: Bass-Serre trees
zbMATH Keywords: HNN extensions
reviewed by: Gregory C. Bell
MaRDI profile type: Publication
full work available at URL: s10711-004-2304-3
OpenAlex ID: W2057890198
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DOI: 10.1007/S10711-004-2304-3
description: scientific article; zbMATH DE number 2145816