Superrigidity for the commensurability group of tree lattices
type: Item
instance of: scholarly article
title: Superrigidity for the commensurability group of tree lattices
zbMATH Open document ID: 0839.22011
DOI: 10.1007/BF02564503
author: Robert J. Zimmer
author: Alexander Lubotzky
author: Shahar Mozes
published in: Commentarii Mathematici Helvetici
publication date: 1996-06-19T00:00:00Z
full work available at URL: https://eudml.org/doc/140353
review text: If \(T = T_k\) is a regular \(k\)-tree \((k \geq 3)\), then its automorphism group \(A\) is a locally compact group which has a number of properties which are analogous to properties enjoyed by simple Lie groups, particularly with respect to lattice subgroups \(\Gamma\). In \textit{H. Bass} and \textit{R. Kulkarni} [J. Am. Math. Soc. 3, 843-902 (1990; Zbl 0734.05052)], it was shown that any two uniform lattices are commensurable up to conjugation, so that the commensurability subgroup \(C\) of such a lattice is uniquely determined by \(A\) up to conjugation. Furthermore \(C\) is dense in \(A\). In this paper, the main result is to show that if \(\rho : C \to \Aut (T')\) defines a minimal action on some tree \(T'\), then either \(\rho\) can be extended to an action of \(A\) or the image of each element of \(\Gamma\) fixes an edge or a vertex so that \(\overline {\rho (\Gamma)}\) is compact. In addition, by exhibiting, for each automorphism \(\rho\) of \(C\), a uniform lattice \(\Gamma\) such that \(\overline {\rho (\Gamma)}\) cannot be compact, it is shown that each automorphism of \(C\) is geometric, i.e. induced by an element of \(A\). The authors offer two proofs of the main result. The first is a graph-theoretic proof, while the second uses methods of ergodic theory. The second proof gives an extension to non-uniform lattices. A number of other consequences of these results is given, together with very clear indications of what remains to be proved and how recent results in this area fit together.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 745323
zbMATH Keywords: regular \(k\)-tree
zbMATH Keywords: automorphism group
zbMATH Keywords: locally compact group
zbMATH Keywords: lattice subgroups
zbMATH Keywords: uniform lattices
zbMATH Keywords: commensurable
zbMATH Keywords: commensurability subgroup
zbMATH Keywords: tree
zbMATH Keywords: ergodic theory
reviewed by: C. Maclachlan
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 745323