Density of the commensurability groups of uniform tree lattices
type: Item
instance of: scholarly article
title: Density of the commensurability groups of uniform tree lattices
zbMATH Open document ID: 0813.20024
author: Ying-Sheng Liu
published in: Journal of Algebra
publication date: 1995-05-23T00:00:00Z
review text: Let \(G\) be a group and \(\Gamma_ 0\), \(\Gamma_ 1\) subgroups of \(G\). The groups \(\Gamma_ 0\) and \(\Gamma_ 1\) are called commensurable if the intersection \(\Gamma_ 0 \cap \Gamma_ 1\) has finite index in both. The commensurator \(C_ G(\Gamma)\) of a subgroup \(\Gamma\) is the group of all \(g \in G\) such that \(\Gamma\) and \(g\Gamma g^{-1}\) are commensurable. If for example \(G\) is a Lie group and \(\Gamma\) an arithmetic subgroup, then the group \(C_ G(\Gamma)\) is essentially the group of rational points of the corresponding arithmetic structure. Now let \(X\) be a uniform tree, i.e. the universal cover of a finite graph. The automorphism group \(G\) of \(X\) is equipped with the compact open topology. Let \(\Gamma\) be a discrete cocompact subgroup of \(G\). The main result of the paper under consideration is that the commensurator \(C_ G(\Gamma)\) of \(\Gamma\) is dense in the topological group \(G\). Under the hypothesis that also the quotient graph \(G\setminus X\) is a tree this was already proven by \textit{H. Bass} and \textit{R. Kulkarni} [J. Am. Math. Soc. 3, No. 4, 843-902 (1990; Zbl 0734.05052)]. For the proof the claim is reduced to the fact that every \(g \in G\) is interpolated on every finite subset of \(X\) by elements of \(C_ G(\Gamma)\). This then is shown by a careful analysis of the combinatorial situation.
reviewed by: Anton Deitmar
Mathematics Subject Classification ID: 20E08
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 589944
zbMATH Keywords: dense commensurator
zbMATH Keywords: commensurable subgroups
zbMATH Keywords: Lie group
zbMATH Keywords: arithmetic subgroup
zbMATH Keywords: group of rational points
zbMATH Keywords: uniform tree
zbMATH Keywords: automorphism group
zbMATH Keywords: discrete cocompact subgroup
MaRDI profile type: Publication
full work available at URL: jabr.1994.1115
OpenAlex ID: W2080049911
DOI: 10.1006/JABR.1994.1115
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description: scientific article; zbMATH DE number 589944