We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semi-simple case. In particular, we extend Mostow's theorem from solvable to amenable groups.
Classification :
22E40
Mots-clés :
Rank of lattices, lattices in Lie groups, finite generation, arithmetic groups
Affiliations des auteurs :
Tsachik Gelander
1
;
Raz Slutsky
1
1
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
Tsachik Gelander; Raz Slutsky. On the Minimal Size of a Generating Set of Lattices in Lie Groups. Journal of Lie Theory, Tome 30 (2020) no. 1, pp. 33-40. http://geodesic.mathdoc.fr/item/JOLT_2020_30_1_a3/
@article{JOLT_2020_30_1_a3,
author = {Tsachik Gelander and Raz Slutsky},
title = {On the {Minimal} {Size} of a {Generating} {Set} of {Lattices} in {Lie} {Groups}},
journal = {Journal of Lie Theory},
pages = {33--40},
year = {2020},
volume = {30},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JOLT_2020_30_1_a3/}
}
TY - JOUR
AU - Tsachik Gelander
AU - Raz Slutsky
TI - On the Minimal Size of a Generating Set of Lattices in Lie Groups
JO - Journal of Lie Theory
PY - 2020
SP - 33
EP - 40
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/item/JOLT_2020_30_1_a3/
ID - JOLT_2020_30_1_a3
ER -
%0 Journal Article
%A Tsachik Gelander
%A Raz Slutsky
%T On the Minimal Size of a Generating Set of Lattices in Lie Groups
%J Journal of Lie Theory
%D 2020
%P 33-40
%V 30
%N 1
%U http://geodesic.mathdoc.fr/item/JOLT_2020_30_1_a3/
%F JOLT_2020_30_1_a3
