Isoperimetric inequalities for lattices in semisimple Lie groups of rank 2
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 \(\text{SL}(3, \mathbb{Z})\).
scientific article; zbMATH DE number 912461
scientific article