Explicit Kazhdan constants for representations of semisimple and arithmetic groups

Yehuda Shalom

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 3, page 833-863
  • ISSN: 0373-0956

Abstract

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Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property ( T ) . For any such group, G , we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice Γ in G are obtained, for a “geometric” generating set of the form Γ ∩ B r , where B r ⊂ G is a ball of radius r , and the dependence of r on Γ is described explicitly. Furthermore, for all rank one Lie groups we derive explicit Kazhdan constants, for any family of representations which admits a spectral gap. Several applications of our methods are discussed as well, among them, an extension of Howe-Moore’s theorem.

How to cite

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Shalom, Yehuda. "Explicit Kazhdan constants for representations of semisimple and arithmetic groups." Annales de l'institut Fourier 50.3 (2000): 833-863. <http://eudml.org/doc/75440>.

@article{Shalom2000,
abstract = {Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $(T)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma $ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r \subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma $ is described explicitly. Furthermore, for all rank one Lie groups we derive explicit Kazhdan constants, for any family of representations which admits a spectral gap. Several applications of our methods are discussed as well, among them, an extension of Howe-Moore’s theorem.},
author = {Shalom, Yehuda},
journal = {Annales de l'institut Fourier},
keywords = {semisimple groups; arithmetic groups; lattices; property ; Kazhdan constants; topological group; Kazhdan set},
language = {eng},
number = {3},
pages = {833-863},
publisher = {Association des Annales de l'Institut Fourier},
title = {Explicit Kazhdan constants for representations of semisimple and arithmetic groups},
url = {http://eudml.org/doc/75440},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Shalom, Yehuda
TI - Explicit Kazhdan constants for representations of semisimple and arithmetic groups
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 3
SP - 833
EP - 863
AB - Consider a simple non-compact algebraic group, over any locally compact non-discrete field, which has Kazhdan’s property $(T)$. For any such group, $G$, we present a Kazhdan set of two elements, and compute its best Kazhdan constant. Then, settling a question raised by Serre and by de la Harpe and Valette, explicit Kazhdan constants for every lattice $\Gamma $ in $G$ are obtained, for a “geometric” generating set of the form $\Gamma \cap B_r$, where $B_r \subset G$ is a ball of radius $r$, and the dependence of $r$ on $\Gamma $ is described explicitly. Furthermore, for all rank one Lie groups we derive explicit Kazhdan constants, for any family of representations which admits a spectral gap. Several applications of our methods are discussed as well, among them, an extension of Howe-Moore’s theorem.
LA - eng
KW - semisimple groups; arithmetic groups; lattices; property ; Kazhdan constants; topological group; Kazhdan set
UR - http://eudml.org/doc/75440
ER -

References

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