Amenability and Kunze-Stein property for groups acting on a tree
type: Item
instance of: scholarly article
title: Amenability and Kunze-Stein property for groups acting on a tree
zbMATH Open document ID: 0671.43003
author: Claudio Nebbia
published in: Pacific Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let X be a locally finite tree and Aut(X) the locally compact group of all isometries of X. It is proved that a closed subgroup G of Aut(X) is amenable if and only if G has one of the following properties: (i) G fixes a vertex; (ii) G leaves invariant an edge; (iii) G fixes an end of X; (iv) G leaves invariant a pair of ends of X. A locally compact group G is said to be a Kunze-Stein group if \(L^ p(G)*L^ 2(G)\subset L^ 2(G)\) for every \(1<p<2\). Let X be a homogeneous tree and G a subgroup of Aut(X) acting transitively on the vertices and on an open subset of the boundary of X. It is shown that G is either amenable or a Kunze-Stein group. The proofs depend on results on \textit{J. Tits} [Essays on topology and related topics, 188-211 (1970; Zbl 0214.513)].
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 43A15
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 05C05
zbMATH DE Number: 4098006
zbMATH Keywords: groups acting on trees
zbMATH Keywords: amenable groups
zbMATH Keywords: locally compact group of
zbMATH Keywords: isometries
zbMATH Keywords: locally finite tree
zbMATH Keywords: Kunze-Stein group
zbMATH Keywords: homogeneous tree
MaRDI profile type: Publication
full work available at URL: pjm.1988.135.371
OpenAlex ID: W2010307579
DOI: 10.2140/PJM.1988.135.371
description: scientific article; zbMATH DE number 4098006