Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold
type: Item
instance of: scholarly article
title: Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold
zbMATH Open document ID: 0599.53036
DOI: 10.1007/BF01450845
author: Marc Burger
author: Viktor Schroeder
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164215
review text: Let X be a simply connected manifold of nonpositive curvature and X(\(\infty)\) be the points at infinity in the sense of Eberlein-O'Neill. In this paper we examine the structure of the group of isometries stabilizing a positive Borel measure on X(\(\infty)\). In this general setting we prove that this group either fixes a point of X(\(\infty)\) or leaves invariant a flat subspace of X. As a corollary we obtain the fact that if the fundamental group of a manifold of finite volume and nonpositive curvature is amenable then this manifold is flat [\textit{A. Avez}, C. R. Acad. Sci., Paris, Sér. A 270, 188-191 (1970; Zbl 0188.264) and \textit{R. J. Zimmer}, Am. J. Math. 105, 1011-1022 (1983; Zbl 0527.57015)]. We also obtain that any amenable subgroup of the fundamental group of a compact manifold of nonpositive curvature is a Bieberbach group, i.e. a finite extension of a finitely generated free abelian group (\textit{Anderson}, to appear in Math. Ann.). We also examine the situation when X is a symmetric space of noncompact type and prove that in this case the stabilizer of a positive Borel measure on X(\(\infty)\) is amenable.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 43A05
zbMATH DE Number: 3967356
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: points at infinity
zbMATH Keywords: group of isometries
zbMATH Keywords: Borel- measure
zbMATH Keywords: finite volume
zbMATH Keywords: amenable subgroup
zbMATH Keywords: Bieberbach group
zbMATH Keywords: symmetric space
MaRDI profile type: Publication
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OpenAlex ID: W1984414998
description: scientific article; zbMATH DE number 3967356