[Les théorèmes d'existence de réseaux associés aux arbres]
Soit X un arbre localement fini, et soit G=Aut(X). Alors G est un groupe localement compact. Par analogie avec les groupes de Lie, Bass et Lubotzky ont conjecturé que G contient des réseaux, c'est-à-dire des sous-groupes discrets dont le quotient porte une mesure invariante finie. Bass et Kulkarni ont montré que G contient des réseaux uniformes si et seulement si G est unimodulaire et G⧹X est fini. Nous décrivons les conditions nécessaires et suffisantes pour que G contienne des réseaux, non seulement uniformes mais aussi non-uniformes, prouvant ainsi complètement les conjectures de Bass et Lubotzky.
Let X be a locally finite tree, and let G=Aut(X). Then G is a locally compact group. In analogy with Lie groups, Bass and Lubotzky conjectured that G contains lattices, that is, discrete subgroups whose quotient carries a finite invariant measure. Bass and Kulkarni showed that G contains uniform lattices if and only if G is unimodular and G⧹X is finite. We describe the necessary and sufficient conditions for G to contain lattices, both uniform and non-uniform, answering the Bass–Lubotzky conjectures in full.
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Lisa Carbone 1
@article{CRMATH_2002__335_3_223_0,
author = {Lisa Carbone},
title = {The tree lattice existence theorems},
journal = {Comptes Rendus. Math\'ematique},
pages = {223--228},
publisher = {Elsevier},
volume = {335},
number = {3},
year = {2002},
doi = {10.1016/S1631-073X(02)02474-3},
language = {en},
}
Lisa Carbone. The tree lattice existence theorems. Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 223-228. doi : 10.1016/S1631-073X(02)02474-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02474-3/
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