On the rigidity of discrete isometry groups of negatively curved spaces
type: Item
instance of: scholarly article
title: On the rigidity of discrete isometry groups of negatively curved spaces
zbMATH Open document ID: 0908.57009
author: Sa'ar Hersonsky
author: Frédéric Paulin
published in: Commentarii Mathematici Helvetici
publication date: 1999-03-10T00:00:00Z
review text: The authors study ergodic properties of discrete groups of isometries of \(\text{CAT}(-1)\)-spaces. The main goal of the paper is to generalize Mostow's rigidity theorem for locally symmetric spaces and Sullivan's rigidity theorem for hyperbolic manifolds to the groups mentioned in the title. Recall that a discrete group of isometries of a \(\text{CAT}(-1)\)-space \(X\) is called group of divergent type (or simply divergent group) if its Poincaré series diverges at the critical exponent. The authors use a generalization of the crossratio due to \textit{J.-P. Otal} [Rev. Mat. Iberoam. 8, No. 3, 441-456 (1992; Zbl 0777.53042)] to define a Möbius map of the ideal boundary \(\partial X\) of \(X\). Namely this is a map of \(\partial X\) which preserves the crossratio. Here is one of the main results of the paper: Theorem A. Let \(X_1, X_2\) be locally complete \(\text{CAT}(-1)\) metric spaces. Let \(\Gamma_1\) and \(\Gamma_2\) be discrete groups of isometries of \(X_1\), \(X_2\), having the same critical exponent. Suppose \(\Gamma_2\) is a divergence group. Let \(\widehat{\phi}:\partial X_1\mapsto \partial X_2\) be a Borel map, equivariant for some morphism \(\Gamma_1\mapsto \Gamma_2\), which is not singular with respect to Patterson-Sullivan measures. Then \(\widehat{\phi}\) is Möbius on the limit set of \(\Gamma_1\). If both \(X_i\;(i=1,2)\) are rank one symmetric spaces, \textit{C. B. Yue} [Invent. Math. 125, No. 1, 75-102 (1996; Zbl 0853.58076)] proved a somewhat stronger result, namely the above theorem is true in this case without requiring that the critical exponents should be equal. The authors provide examples of non-trivial applications of their theorem to \(\text{CAT}(-1)\) spaces different than rank one symmetric spaces and trees. Another important result of the paper is related to \textit{J. P. Otal}'s rigidity theorem for marked length spectra of closed surfaces [Ann. Math., II. Ser. 131, No. 1, 151-162 (1990; Zbl 0699.58018)]. The authors obtain an analogo…
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 51M10
Mathematics Subject Classification ID: 51E24
zbMATH DE Number: 1121235
zbMATH Keywords: Mostow's rigidity theorem
zbMATH Keywords: Sullivan's rigidity theorem
zbMATH Keywords: \(\text{CAT}(-1)\)-space
zbMATH Keywords: Möbius map
reviewed by: Leonid Potyagailo
MaRDI profile type: Publication
full work available at URL: s000140050022
OpenAlex ID: W2035100216
Wikidata QID: Q125096408
DOI: 10.1007/S000140050022
Recommended article: Superrigidity for homomorphisms into isometry groups of \(\text{CAT}(-1)\) spaces
Recommended article: Ergodic theory and rigidity on the symmetric space of non-compact type
Recommended article: Ergodicité et équidistribution en courbure négative
Recommended article: Q3119369
Recommended article: Equidistribution, ergodicity and irreducibility in CAT(\(-1\)) spaces
Recommended article: On the geodesic flow on CAT(0) spaces
Recommended article: Q4014847
Recommended article: Tukia's isomorphism theorem in CAT(–1) spaces
Recommended article: Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces
Recommended article: Cross ratios on boundaries of symmetric spaces and euclidean buildings
description: scientific article; zbMATH DE number 1121235