The tree lattice existence theorems
type: Item
instance of: scholarly article
title: The tree lattice existence theorems
zbMATH Open document ID: 1025.20013
DOI: 10.1016/S1631-073X(02)02474-3
author: Lisa Carbone
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-11-17T00:00:00Z
review text: Let \(X\) be a locally finite tree and \(G=\Aut(X)\). Then \(G\) is a locally compact group. Let \(\mu\) be a left invariant Haar measure on \(G\). A discrete subgroup \(\Gamma\) of \(G\) is a \(G\)-lattice, if \(\mu(\Gamma\backslash G)\) is finite. It is uniform if \(\Gamma\backslash G\) is compact, non-uniform otherwise. In analogy with Lie groups, Bass and Lubotzky conjectured that \(G=\Aut(X)\) contains lattices. In [\textit{H. Bass} and \textit{R. Kulkarni}, J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)] it was shown that \(G\) contains a uniform lattice if and only if it is unimodular and \(G\backslash X\) is finite. The paper under review presents theorems that prove this conjecture in full, i.e. it presents necessary and sufficient conditions for the existence of both uniform and non-uniform \(G\)-lattices. The proofs of these theorems appear in [(*) \textit{H. Bass}, \textit{L. Carbone} and \textit{G. Rosenberg}, Appendix [BCR] in H. Bass and A. Lubotzky, Tree Lattices, Progr. Math. 176, Birkhäuser, Boston (2001; Zbl 1053.20026); (**) \textit{L. Carbone}, Mem. Am. Math. Soc. 724 (2001; Zbl 0983.05002); (***) \textit{L. Carbone} [preprint]; (****) \textit{L. Carbone} and \textit{D. Clark}, Commun. Algebra 30, No. 4, 1853-1886 (2002; Zbl 1004.22005); (*****) \textit{L. Carbone} and \textit{G. Rosenberg}, Geom. Dedicata 98, 161-188 (2003; Zbl 1055.20021)]. The construction of the \(G\)-lattices above relies on the construction of \(X\)-lattices. A subgroup \(\Gamma\subset G\) is an \(X\)-lattice if \(\text{Vol}(\Gamma\backslash \backslash X)\) is finite. It is known [\textit{H. Bass} and \textit{A. Lubotzky}, Tree lattices (Progr. Math. 176, Birkäuser, Boston) (2001; Zbl 1053.20026)] that \(\Gamma\) is an \(X\)-lattice if and only if \(\Gamma\) is a \(G\)-lattice and \(\mu(G\backslash \backslash X)<\infty\). Theorem 0.2 [(*)] of the paper under review states that \(G\) contains an \(X\)-lattice \(\Gamma\) if and only if \(G\) is unimodular and \(\…
reviewed by: L. Lifschitz
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1854342
zbMATH Keywords: locally finite trees
zbMATH Keywords: locally compact groups
zbMATH Keywords: left invariant Haar measures
zbMATH Keywords: discrete subgroups
zbMATH Keywords: lattices
zbMATH Keywords: uniform lattices
zbMATH Keywords: non-uniform \(G\)-lattices
MaRDI profile type: Publication
cites work: Covering theory for graphs of groups
cites work: Uniform Tree Lattices
cites work: Non-uniform lattices on uniform trees
cites work: LATTICES ON PARABOLIC TREES
cites work: Lattices on nonuniform trees.
cites work: Infinite towers of tree lattices
cites work: Lattices in rank one Lie groups over local fields
cites work: Q4867087
cites work: Q3341041
description: scientific article; zbMATH DE number 1854342