Journal of Lie Theory 30 (2020), No. 1, 033--040
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 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.
Keywords: Rank of lattices, lattices in Lie groups, finite generation, arithmetic groups.
MSC: 22E40
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