Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
type: Item
instance of: scholarly article
title: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
zbMATH Open document ID: 0678.53042
DOI: 10.2307/1971484
author: Pierre Pansu
published in: Annals of Mathematics. Second Series
publication date: 1989-01-01T00:00:00Z
review text: A continuous map \(f\) between metric spaces \(X_1\) and \(X_2\) is a quasiisometry if there are positive constants \(C\) and \(L\) such that \(-C+(1/L)d(x,y)\le d(fx,fy)\le C+Ld(x,y)\) for all points \(x,y\) in \(X_ 1\). Now let \(\tilde M\) denote a symmetric space of rank 1 (i.e. strictly negative sectional curvature) and let \(\tilde M(\infty)\) denote the boundary sphere of \(\tilde M\). Morse showed essentially that any quasiisometry \(f: \tilde M\to \tilde M\) extends to a continuous map \(\bar f: \tilde M(\infty)\to \tilde M(\infty)\). Mostow and the author showed that \(\bar f\) is in fact quasiconformal with respect to a non-Riemannian Carnot-Carathéodory metric \(\delta\) on \(\tilde M(\infty)\). The metric \(\delta\) actually depends on the choice of a point \(x\) in \(\tilde M(\infty)\) and a horosphere \(H\) at \(x\), but quasiconformality and 1-quasiconformality for a function \(f: \tilde M(\infty)\to \tilde M(\infty)\) are independent of \(x\) and \(H\). In this paper the author proves the following main theorem. Let \(\tilde M\) be the \(n\)-dimensional quaternionic hyperbolic space or the Cayley plane, and let \(f: \tilde M\to \tilde M\) be a quasiisometry. Then there is an isometry \(g: \tilde M\to \tilde M\) such that \(d(fp,gp)\le C\) for some \(C>0\) and all points \(p\) in \(\tilde M\). We sketch a proof. Given a point \(x\) in \(\tilde M(\infty)\) one may identify \(\tilde M(\infty)-\{x\}\) with a simply connected 2-step nilpotent group \(N\) of isometries of \(\tilde M\) that acts simply transitively on horospheres at \(x\). The point \(x\) also determines a derivation \(\alpha\) of the Lie algebra of \(N\). If \(f: \tilde M\to \tilde M\) is a quasiisometry, then for almost all points \(n\) of \(N\) the extension \(\bar f: \tilde M(\infty)\to \tilde M(\infty)\) induces a ``\(\delta\)-differential'' \(df_n: N\to N\), which is an automorphism of \(N\) that commutes with \(\alpha\). If \(\tilde M\) is a quaternionic hyperbolic space or the Cayl…
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 4111501
zbMATH Keywords: Mostow rigidity
zbMATH Keywords: quasiisometry
zbMATH Keywords: symmetric space of rank 1
zbMATH Keywords: Carnot- Carathéodory metric
zbMATH Keywords: quasiconformality
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971484
OpenAlex ID: W2314781788
Wikidata QID: Q56850582
description: scientific article; zbMATH DE number 4111501