A simple proof of a theorem of Whyte.
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 \(BS(2,3).\)
scientific article; zbMATH DE number 2145816