Lattices on nonuniform trees.
type: Item
instance of: scholarly article
title: Lattices on nonuniform trees.
zbMATH Open document ID: 1055.20021
DOI: 10.1023/A:1024053813643
author: Lisa Carbone
author: Gabriel Rosenberg
published in: Geometriae Dedicata
publication date: 2003-08-06T00:00:00Z
review text: Let \(X\) be a locally finite tree, then \(G=\Aut(X)\) is a locally compact group with compact open vertex stabilizers \(G_x\), \(x\in VX\). We say \(X\) is a `uniform' tree, if \(G\setminus X\) is finite, otherwise \(X\) is a `non-uniform' tree. A subgroup \(\Gamma\subset G\) is `discrete' if \(\Gamma_x\) is finite for some, and hence every vertex \(x\in VX\). Let \(\mu\) denote a (left) Haar measure on \(G\). A discrete subgroup \(\Gamma\subset G\) is a `\(G\)-lattice', if \(\mu(\Gamma\setminus G)\) is finite. We say \(\Gamma\) is a `uniform' \(G\)-lattice if \(\Gamma\setminus G\) is compact and `non-uniform' otherwise. A discrete subgroup \(\Gamma\subset G\) is called an `\(X\)-lattice' if \[ \text{Vol}(\Gamma\backslash\backslash X)=\sum_{x\in V(\Gamma\setminus X)}\tfrac 1{|\Gamma_x|} \] is finite, a `uniform' \(X\)-lattice if \(\Gamma\setminus X\) is a finite graph and `non-uniform' \(X\)-lattice if \(\Gamma\setminus X\) is infinite. It was shown by \textit{H. Bass} and \textit{R. Kulkarni} [J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)] that \(G\) contains a uniform \(X\)-lattice if and only if \(X\) is the universal cover of a finite connected graph or, equivalently, \(G\) is unimodular and \(G\setminus X\) is finite. In [\textit{H. Bass}, \textit{L. Carbone} and \textit{G. Rosenberg}, Appendix [BCR] in \textit{H. Bass} and \textit{A. Lubotzky}, Tree Lattices (Progr. Math. 176, Birkhäuser, Boston) (2000; Zbl 1053.20026)] it was shown that \(G\) contains an \(X\)-lattice \(\Gamma\) if and only if \(G\) is unimodular and \[ \mu(G\backslash\backslash X)=\sum_{x\in V(G\setminus X)}\tfrac 1{\mu(G_x)}<\infty. \] In particular, if \(G\) is unimodular, \(\mu(G\backslash\backslash X)<\infty\) and \(G\setminus X\) is infinite, \(G\) contains a necessarily non-uniform \(X\)-lattice \(\Gamma\). However, \(\Gamma\) turned out to be a uniform \(G\)-lattice. The main result of the present paper states that given a locally finite tree \(X\) with more than one end…
reviewed by: L. Lifschitz
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 22F50
zbMATH DE Number: 1960061
zbMATH Keywords: lattices
zbMATH Keywords: locally finite trees
zbMATH Keywords: discrete groups
zbMATH Keywords: volumes
zbMATH Keywords: locally compact groups
zbMATH Keywords: numbers of ends
MaRDI profile type: Publication
full work available at URL: a:1024053813643
OpenAlex ID: W1515878136
Recommended article: The tree lattice existence theorems
Recommended article: Tree lattices. With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits.
Recommended article: Q4495678
Recommended article: Non-minimal tree actions and the existence of non-uniform tree lattices
Recommended article: Tree lattice subgroups.
Recommended article: Non-uniform lattices on uniform trees
Recommended article: Infinite towers of tree lattices
Recommended article: Superrigidity for the commensurability group of tree lattices
Recommended article: Indicability, residual finiteness, and simple subquotients of groups acting on trees
Recommended article: Counting overlattices in automorphism groups of trees.
description: scientific article; zbMATH DE number 1960061