Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2
type: Item
instance of: scholarly article
title: Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2
zbMATH Open document ID: 0856.22013
DOI: 10.1007/BF02249261
author: Enrico Leuzinger
author: Christophe Pittet
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-02-03T00:00:00Z
full work available at URL: https://eudml.org/doc/58234
review text: The main result of this paper is the following Theorem 1.1. Let \(G\) be a connected semisimple Lie group of real rank 2 with finite center. Let \(\Gamma\) be a non-uniform irreducible lattice in \(G\). Then \(\delta_\Gamma(n)\simeq \exp(n)\). -- The paper is organized as follows. \S 2 contains a sketch of the proof of Theorem 1.1. \S 3 contains a result of Gromov on filling functions of certain solvable groups. In \S 4 the authors describe configurations of rank 2 root systems with a ray starting from the origin. In \S 5 the authors relate the irreducibility of \(\Gamma\) to ``cross-like configurations'' of root systems. \S 6 contains two results on the structure of algebraic groups together with their geometric interpretations. They both are essential for the proof. \S 7 is the proof of the theorem. \S 8 is an appendix which illustrates the general case by the example of \(\text{SL}(3, \mathbb{Z})\).
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 912461
zbMATH Keywords: cross-like configurations
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: lattice
zbMATH Keywords: filling functions
zbMATH Keywords: solvable groups
zbMATH Keywords: root systems
zbMATH Keywords: algebraic groups
Wikidata QID: Q115392660
reviewed by: L. A. Kurdachenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 912461