
Journal of Lie Theory 30 (2020), No. 1, 033040 Copyright Heldermann Verlag 2020 On the Minimal Size of a Generating Set of Lattices in Lie Groups Tsachik Gelander Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel Tsachik.gelander@gmail.com Raz Slutsky Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel Razslo@gmail.com 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 covolume of the projection of the lattice to the semisimple part of the group. This was proved by Gelander for semisimple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semisimple case. In particular, we extend Mostow's theorem from solvable to amenable groups. Keywords: Rank of lattices, lattices in Lie groups, finite generation, arithmetic groups. MSC: 22E40 [ Fulltextpdf (104 KB)] for subscribers only. 