We introduce a family of rigid hyperbolic 3-manifolds of infinite volume with possibly infinitely many ends: the treelike manifolds. These manifolds generalize a family of constructive non compact surfaces – the equational surfaces – for which the homeomorphism problem is decidable. The proof of rigidity relies firstly on Thurston's theorem of compactness of the Teichmüller space of acylindrical compact 3-manifolds, and secondly, on Sullivan's rigidity theorem.
Nous introduisons une famille de 3-variétés hyperboliques rigides de volume infini à nombre de bouts infini : les variétés arborescentes. Ces variétés généralisent une famille de surfaces non compactes constructives – les surfaces équationnelles – pour lesquelles le problème de l'homéomorphisme est décidable. La démonstration de rigidité s'appuie sur, premièrement, le théorème de Thurston de compacité de l'espace de Teichmüller des 3-variétés compactes acylindriques, et deuxièmement, le théorème de rigidité de Sullivan.
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Olivier Ly 1
@article{CRMATH_2003__337_7_457_0,
author = {Olivier Ly},
title = {Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume},
journal = {Comptes Rendus. Math\'ematique},
pages = {457--460},
publisher = {Elsevier},
volume = {337},
number = {7},
year = {2003},
doi = {10.1016/j.crma.2003.08.005},
language = {en},
}
Olivier Ly. Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume. Comptes Rendus. Mathématique, Volume 337 (2003) no. 7, pp. 457-460. doi : 10.1016/j.crma.2003.08.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.08.005/
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