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.
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.
Accepted:
Published online:
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/
[1] Covering theory for graphs of groups, J. Pure Appl. Algebra, Volume 89 (1993), pp. 3-47
[2] The existence theorem for tree lattices, Appendix [BCR], Tree Lattices, 176, Birkhäuser, Boston, 2000 (in: H. Bass, A. Lubotzky, Progr. Math.)
[3] Uniform tree lattices, J. Amer. Math. Soc., Volume 3 (1990) no. 4
[4] Tree Lattices, Progr. Math., 176, Birkhäuser, Boston, 2000
[5] A discreteness criterion for certain tree automorphism groups, Appendix [BT], Tree Lattices, 176, Birkhäuser, Boston, 2000 (in: H. Bass, A. Lubotzky, Progr. Math.)
[6] Non-uniform lattices on uniform trees, Mem. Amer. Math. Soc., Volume 152 (2001) no. 724
[7] L. Carbone, Non-minimal tree actions and the existence of non-uniform tree lattices, Preprint, 2002
[8] Lattices on parabolic trees, Comm. Algebra, Volume 30 (2002) no. 4
[9] L. Carbone, G. Rosenberg, Lattices on non-uniform trees, Geom. Dedicate (2002), to appear
[10] Infinite towers of tree lattices, Math. Res. Lett., Volume 8 (2001), pp. 1-10
[11] L. Carbone, G. Rosenberg, Infinite towers of non-uniform tree lattices (2002), in preparation
[12] A proof of Selberg's hypothesis, Mat. Sb. (N.S.), Volume 75 (1968) no. 117, pp. 163-168 (in Russian)
[13] Lattices in rank one Lie groups over local fields, Geom. Funct. Anal., Volume 1 (1991) no. 4, pp. 405-431
[14] Tree lattices and lattices in Lie groups (A. Duncan; N. Gilbert; J. Howie, eds.), Combinatorial and Geometric Group Theory, LMS Lecture Note Series, 204, Cambridge University Press, 1995, pp. 217-232
[15] G. Rosenberg, Towers and covolumes of tree lattices, Ph.D. Thesis, Columbia University, 2000
[16] Trees, Springer-Verlag, Berlin, 1980 (Translated from the French by J. Stilwell)
Cited by Sources:
Comments - Policy