A fixed point theorem of discrete group actions on Riemannian manifolds
type: Item
instance of: scholarly article
title: A fixed point theorem of discrete group actions on Riemannian manifolds
zbMATH Open document ID: 0951.58020
author: M.-T. Wang
published in: Journal of Differential Geometry
publication date: 1999-11-23T00:00:00Z
review text: A group \(\Gamma\) has property \((F)\) iff any isometric action of \(\Gamma\) on a manifold \(N\) of nonpositive sectional curvature has a fixed point in the compactification \(\overline N\). The author proves the following theorem: If \(\Sigma\) is a finite simplicial complex with an admissible weight \(c\) such that \(\lambda_{1,\text{loc}}(\Sigma,c)>{1\over 2}\), then the fundamental group of~\(\Sigma\) has property \((F)\). Here, an admissible weight is a metric on~\(C^*(\Sigma)\) of a certain type, and the condition~\(\lambda_{1,\text{loc}}(\Sigma,c)>{1\over 2}\) refers to the Laplacian on \(0\)-forms on the links of vertices in \(\Sigma\). The proof uses harmonic maps from simplical complexes to manifolds. The theorem above implies the superrigidity theorem of \textit{G. A. Margulis} [`Discrete subgroups of semisimple Lie groups', Springer (1991; Zbl 0732.22008)] for cocompact lattices in \(p\)-adic groups of rank greater than one, at least if the residue field is sufficiently large.
reviewed by: Sebastian Goette
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 1368005
zbMATH Keywords: property(F)
zbMATH Keywords: buildings
zbMATH Keywords: harmonic maps
zbMATH Keywords: superrigidity
zbMATH Keywords: nonpositive curvature
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214461170
OpenAlex ID: W1556043334
Wikidata QID: Q115173787
DOI: 10.4310/JDG/1214461170
description: scientific article; zbMATH DE number 1368005