Tree lattices. With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits.
type: Item
instance of: scholarly article
title: Tree lattices. With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits.
zbMATH Open document ID: 1053.20026
author: Hyman Bass
author: Alexander Lubotzky
published in: Progress in Mathematics
publication date: 2001-01-07T00:00:00Z
review text: Let \(X\) be a locally finite tree with the vertex set \(VX\), and let for the vertex \(x\in VX\) the group \(G_x\) be its stabilizer in the locally compact group \(G=\Aut(X)\). A subgroup \(\Gamma\leq G\) is said to be discrete if \(\Gamma_x\) is finite for some, and hence every vertex \(x\in VX\). Then we can define the sum \(\text{Vol}(\Gamma\backslash\backslash X)=\sum_{x\in\Gamma\setminus VX}|\Gamma_x|^{-1}\). \(\Gamma\) is defined to be an \(X\)-lattice if \(\text{Vol}(\Gamma\backslash\backslash X)<\infty\), and it is defined to be a uniform \(X\)-lattice if \(\Gamma\setminus X\) is a finite graph. If \(G\setminus X\) is finite, these are equivalent to \(\Gamma\) being a lattice and uniform lattice, respectively. The current investigation studies these tree lattices, and it in some sense is a sequel to the work of \textit{H. Bass} and \textit{R. Kulkarni} [J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)], where the study of the uniform tree lattices was initiated. Here the work is focused much more on the non-uniform case which is more complex. This approach is the generalization of the following. A non-compact simple Lie group \(H\) in most cases can be realized as an essentially full isometry group of some contractible metric space \(X\). The geometric action is a natural way to study \(H\) and its discrete subgroups, in particular its lattices. Chapter~1 provides a brief summary for basic facts about lattices in locally compact groups. Chapter~2 reviews some basic material on graphs of groups and edge-indexed graphs. Chapter~3 presents the notion of tree lattices and their basic properties. Chapter~4 shows that if \(X\) is a locally finite tree and \(\Gamma\) is a non-uniform \(X\)-lattice, then, even when \(X\) is regular of degree \(\geq 3\), \(\text{Vol}(\Gamma\backslash\backslash X)\) can take any positive real value, that \(\Gamma\setminus X\) can have any conceivable number of ``cusps'' (in some appropriate sense). Chapter~5 reviews the pro…
reviewed by: Vahagn H. Mikaelian
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 1548758
zbMATH Keywords: tree lattices
zbMATH Keywords: groups acting on trees
zbMATH Keywords: free products of groups
zbMATH Keywords: HNN-extensions
zbMATH Keywords: geometric group theory
zbMATH Keywords: discrete subgroups of Lie groups
zbMATH Keywords: locally compact groups
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1548758